CONDITIONS FOR THE EXISTENCE OF SBR MEASURES

FOR “ALMOST ANOSOV” DIFFEOMORPHISMS

Huyi Hu

Abstract. A diffeomorphism f of a compact manifold M is called “almost Anosov” if it is uniformly hyperbolic away from a finite set of points. We show that under some nondegeneracy condition, every almost Anosov diffeomorphism admits an invariant measure μ that has absolutely continuous conditional measures on unstable manifolds. The measure μ is either finite or infinite, and is called SBR measure or infinite SBR measure respectively. Therefore, \frac{1}{n} ∑_{i=0}^{n−1} δ_{f^{i}x} tends to either an SBR measure or δ_{p} for almost every x with respect to Lebesgue measure. (δ_{x} is the Dirac measure at x.) For each case, we give sufficient conditions by using coefficients of the third order terms in the Taylor expansion of f at p.

0. Introduction

In this paper we study the existence of SBR measures of two dimensional diffeomorphisms that are hyperbolic everywhere except at finite points. We remark that it is easy to see that our methods extend to the situation in which hyperbolicity fails at only finitely many periodic points.

If f : M → M is a C^{2} Anosov diffeomorphism of a compact connected Riemannian manifold, then a result of Sinai (See e.g. [S]) says that f admits an invariant Borel probability measure μ with the property that μ has absolutely continuous conditional measures on unstable manifolds. With respect to this measure, Lebesgue almost every point is generic. That is, if ϕ : M → IR is a continuous function, then for Lebesgue almost every x ∈ M,

lim_{n→∞} \frac{1}{n} ∑_{i=0}^{n−1} ϕ(f^{i}x) → ∫ ϕdμ.  (0.1)

This result, as well as some other properties of μ, has been extended to Axiom A attractors by Bowen, Ruelle, etc. (See e.g. [B].) In this article we will refer to an

1991 Mathematics Subject Classification. Primary 58F11, 58F15; Secondary 28d05. Key words and phrases. Almost Anosov diffeomorphism, SBR measure, Infinite SBR measure,

Local Hölder condition.

The author of this work was supported by NSF under grants DMS-8802593 and DMS-9116391, and by DOE (Office of Scientific Computing).

Typeset by AMS-TEX

invariant measure having absolutely continuous conditional measures on unstable manifolds as a Sinai-Bowen-Ruelle measure or an SBR measure. Due to the works of Oseledec, Pesin, Ledrappier, Young and others on nonuniformly hyperbolic set, the notion of SBR measure is extended to a more general setting (see [O], [P1] and [LS]), some properties of SBR measure are obtained (e.g. [L] and [LY]). Furthermore, a few examples of SBR measure outside Axiom A systems are studied (see e.g. [BY],

[P2] and [C]).

This paper is motivated by the following question: Does a system f : M → M admit an SBR measure if hyperbolicity fails at only one fixed point p? We confine our topic to two dimensional cases. Suppose in this system one of the eigenvalues of the derivative Df_{p} is larger than 1 and the other is equal to 1. The map is in fact uniformly expanding along unstable directions. It is easy to see that the system admits an SBR measure μ, because the arguments on bounded distortion estimates are standard and the push-forward method for invariant measure works. Consequently, the set of generic points with respect to μ has full Lebesgue measure. If the larger eigenvalue is equal to 1 and the smaller one is less than 1, then the results in [HY] indicate that it does not admit SBR measures, and the limit in the left hand side in (0.1) is ϕ(p) at Lebesgue almost every point. It is also found that this system admits an infinite measure that has absolutely continuous conditional measures on weak unstable manifolds. Here we will refer to this measure as an infinite SBR measure (see §1 for precise meaning). In this paper, we investigate the case that the derivative Df_{p} is identity. Our results show that, under some nondegeneracy conditions, f admits either an SBR measure or an infinite SBR measure (Theorem A), and both cases do occur (Theorem B).

The phenomena in this case are quite different with that we mentioned above, and some interesting things happen. For example, the decomposition of the tangent spaces into T_{x}M = E_{x}^{u} ⊕ E_{x}^{s} is discontinuous at the fixed point p (see Remark 4.3). Consequently we cannot expect the Hölder condition for E_{x}^{u} and the Lipschitz condition for W^{s}-foliation, which are used for bounded distortion and Lebesgue genericity in [HY]. However, we find out that E_{x}^{u} satisfies Hölder condition away from p and the W^{s}-foliation remains absolute continuous despite the discontinuity of E_{x}^{u} and E_{x}^{s}. On the other hand, the failure of Lipschitzness of the W^{s}-foliation makes both SBR measure and infinite SBR measure possible. Notice that whether the measure is finite or infinite depends on whether the area of the sets P_{n} = {x ∈ P : f^{i}x ∈ P, i = 0, 1, ⋯, n} can be summed up or not for any rectangle P containing p in its interior. For a normal Anosov system or a system considered in [HY], because of the Lipschitzness of W^{s}-foliation this solely depends on how fast a point x in W^{s}(p)\{p} approachs to p under backwards iteration. But in our case it depends on how much a stable manifold W^{s}(x) bends to W^{s}(p) for x near p.

The proofs of the theorems involve detailed analysis of the dynamics near the “hyperbolic type” indifferent fixed point, which hasn’t been done before. We find out a similarity, i.e. the behavior of f^{n^{2}} at \frac{x}{n} is about the same as f at x but in a smaller scale. We prove the existence and differentiability of stable and unstable manifolds for the fixed point p. We use eigenspaces as coordinates systems to study properties of the splittings E_{x}^{u}⊕E_{x}^{s}, and use the “local Hölder condition” away from p to avoid the problem cuased by discontinuity of the splittings at p, the latter is crucial for the estimates of bounded distortion.

SBR measures on the Hénon attractors, whose existence is found by Benedicks and L.-S. Young ([BY]), are notable examples for non-uniformly hyperbolic systems. The Hénon attractors involve nontrivial interchanges between stable and unstable directions. There exist some areas in which the maps contract very severely along the directions corresponding to positive Lyapunov exponents. However, the orbits jump out immediately and then stay outside the areas for a while so that expansion can be recovered. In our case the stable and unstable directions are bounded away from each other and the map is always expanding in unstable directions. But near the fixed point p the expansion is very weak and the orbits spend a very long time there. This causes non-uniform hyperbolicity. The speed of expansion may or may not be able to keep exponentially fast eventually, depending on the local behaviors of the maps (see Remark 1.5 and Theorem B).

Acknowledgment. Part of the work is done while I was in the Department of Mathematics, University of Arizona. It is my pleasure to thank my advisor Professor Lai-Sang Young for introducing me this topic and giving valuable advices. I also wish to think Professor Marek Rychlik for his helpful conversation.

1. Definitions and Statement of Results

Let M be a C^{∞} two-dimensional compact Riemannian manifold without boundary and let m denote the Riemannian measure on M. Let f ∈ Diff^{4}(M), the set of C^{4} diffeomorphisms.

Definition 1. f ∈ Diff^{4}(M) is said to be an almost Anosov diffeomorphism, if there exist two continuous families of cones x → C_{x}^{u}, C_{x}^{s} such that except for a finite set S,

i) Df_{x}C_{x}^{u} ⊆ C_{fx}^{u} and Df_{x}C_{x}^{s} ⊇ C_{fx}^{s},
ii) |Df_{x}v| > |v| ∀v ∈ C_{x}^{u} and |Df_{x}v| < |v| ∀v ∈ C_{x}^{s}.

Remark 1.1. We may assume S is an invariant set. Moreover, by considering f^{n} instead of f we may also assume fp = p ∀p ∈ S.

Remark 1.2. By the continuity, it is easy to see that if p ∈ S, then i) Df_{p}C_{p}^{u} ⊆ C_{p}^{u} and Df_{p}C_{p}^{s} ⊇ C_{p}^{s}, ii) |Df_{p}v| ≥ |v| ∀v ∈ C_{p}^{u} and |Df_{p}v| ≤ |v| ∀v ∈ C_{p}^{s}.

For Γ ⊂ M and r > 0, we denote B(Γ, r) = {y ∈ M : d(y, Γ) ≤ r}.

The ratio of |Df_{x}v| to |v| may tend to 1 if x → S and v ∈ C_{x}^{u} or C_{x}^{s}. We need some condition to control the speed of the ratio.

Definition 2. An almost Anosov diffeomorphism f is said to be nondegenerate (up to the third order ), if there exist constants r_{0} > 0 and κ^{u}, κ^{s} > 0 such that for all x ∈ B(S, r_{0}),

\begin{matrix}|Df_{x}v| ≥ (1 + κ^{u}d(x, S)^{2})|v|  ∀v ∈ C_{x}^{u}, \\ |Df_{x}v| ≤ (1 − κ^{s}d(x, S)^{2})|v|  ∀v ∈ C_{x}^{s}. \end{matrix}  (1.1)

Remark 1.3. If f is an almost Anosov diffeomorphism, then for any constant r > 0, there exist constants 0 < K^{s} < 1 < K^{u}, depending on r, such that for all x ∉ B(S, r),

\begin{matrix}|Df_{x}v| ≥ K^{u}|v|  ∀v ∈ C_{x}^{u}, \\ |Df_{x}v| ≤ K^{s}|v|  ∀v ∈ C_{x}^{s}. \end{matrix}  (1.2)

The following paragraphs concerning the definition of SBR measures can be seen in [LS]. We state here for the convenience of the reader.

Let ξ be a measurable partition of a measure space X, and let ν be a probability measure on X. Then there is a family of probability measures {ν_{x}^{ξ} : x ∈ X} with ν_{x}^{ξ}(ξ(x)) = 1, such that for every measurable set B ⊂ X, x → ν_{x}^{ξ}(B) is measurable and

νB = ∫_{X} ν_{x}^{ξ}(B)dν(x).

The family {ν_{x}^{ξ}} is called a canonical system of conditional measures for ν and ξ. (For reference, see e.g. [R].)

Suppose now that f : (M, μ) → (M, μ) has positive Lyapunov exponents almost everywhere. Then for μ -a.e. x, the unstable manifold W^{u}(x) exists and is an immersed submanifold of M (see [P1]). A measurable partition ξ of M is said to be subordinate to unstable manifolds if for μ -a.e. x, ξ(x) ⊂ W^{u}(x) and contains an open neighborhood of x in W^{u}(x). Let m_{x}^{u} denote the Riemannian measure induced on W^{u}(x). We say that μ has absolutely continuous conditional measures on unstable manifolds if for every measurable partition ξ that is subordinate to unstable manifolds, μ_{x}^{ξ} is absolutely continuous with respect to m_{x}^{u} (written μ_{x}^{ξ} < < m_{x}^{u}) for μ -a.e. x ∈ M. (For more details, see [LS].)

Definition 3. An f-invariant Borel probability measure μ on M is called an SBR measure for f : M → M if

i) (f, μ) has positive Lyapunov exponents almost everywhere; ii) μ has absolutely continuous conditional measures on unstable manifolds.

Let Γ be a subset of M. We denote by f_{Γ} the first return map on Γ, i.e. f_{Γ}x = f^{τ(x)}(x) ∀x ∈ Γ, where τ(x) is the smallest positive integer such that f^{τ(x)}(x) ∈ Γ. We also denote by μ_{Γ} the normalization of μ|_{Γ} as μΓ < ∞.

Note that the notion of absolutely continuous conditional measures on unstable manifolds makes sense even f is a piecewise diffeomorphism.

Definition 4. An f-invariant Borel measure μ on M is called an infinite SBR measure, if μM = ∞ and for any open set U ⊃ S,

i) μ(M\U) < ∞,

ii) (f_{M\U}, μ_{M\U}) has positive Lyapunov exponents almost everywhere, and μ_{M\U} has absolutely continuous conditional measures on unstable manifolds of f.

Remark 1.4. In this paper, the term “SBR measure” without any qualifications will always be reserved for probability measures.

Remember our assumption that M is a C^{∞} two-dimensional compact Riemannian manifold, m is the Riemannian measure on M and f ∈ Diff^{4}(M).

Theorem A. Every topologically transitive nondegenerate almost Anosov diffeomorphism f on M has either an SBR measure or an infinite SBR measure.

The nondegeneracy condition in the theorem is only for technical reasons. The author believes that the result would remain true under some weaker conditions.

Corollary. Let f be as in Theorem A.

i) If f admits an SBR measure μ, then for any continuous function ϕ : M → IR,

lim_{n→∞} \frac{1}{n} ∑_{i=0}^{n−1} ϕ(f^{i}x) = ∫ ϕdμ m -a.e. x ∈ M.

ii) If f admits an infinite SBR measure μ, then for any open neighborhood U of S,

lim_{n→∞} \frac{1}{n}#\{k : f^{k}x ∈ U, 0 ≤ k ≤ n − 1\} = 1 m -a.e. x ∈ M.

Remark 1.5. In case i) Lebesgue almost every point has a positive Lyapunov exponent because of the definition of the SBR measure. On the contrary, it is evident by the corollary that in case ii) there is no positive Lyapunov exponent at Lebesgue almost every point in M.

In this paper we only consider the case that S contains a single point, i.e. S = {p}. It is not difficult for the reader to adjust the proof to the case that S contains more than one point.

If S = {p}, then the part ii) of the above corollary is equivalent to the following.

ii′) If f admits an infinite SBR measure, then for any continuous function ϕ : M → IR,

lim_{n→∞} \frac{1}{n} ∑_{i=0}^{n−1} ϕ(f^{i}x) = ϕ(p) m -a.e. x ∈ M.

By Remark 1.3, we see that Df_{p} have two eigenvectors. As we mentioned in §0, it is known that Theorem A holds if at least one of eigenvalues of Df_{p} is not equal to 1. So for proving Theorem A, we only need consider the case that Df_{p} = id.

For further analysis we need the following technical assumptions. Denote

\begin{aligned}W_{ϵ}^{u}(p) & = \{y ∈ M : d(f^{−n}y, p) ≤ ϵ ∀n ≥ 0\}, \\ W_{ϵ}^{s}(p) & = \{y ∈ M : d(f^{n} y, p) ≤ ϵ ∀n ≥ 0\}. \end{aligned}

We will prove in §4 that both are differentiable curves. To state and prove Theorem B we give the following.

Assumption A. W_{ϵ}^{u}(p) and W_{ϵ}^{s}(p) are C^{4} curves.

Note Df_{p} = id. We will show D^{2}f_{p} = 0 (Proposition 2.1). Thus we can take a suitable coordinate system such that in some neighborhood of p, f can be expressed as

f(x, y) = ( x(1 + ϕ(x, y)), y(1 − ψ(x, y)) ),  (1.3)

where (x, y) ∈ IR^{2} and

ϕ(x, y) = a_{0}x^{2} + a_{1}xy + a_{2}y^{2} + O(|(x, y)|^{3}),
ψ(x, y) = b_{0}x^{2} + b_{1}xy + b_{2}y^{2} + O(|(x, y)|^{3}).  (1.4)

In this circumstance we have the following.

Theorem B. Let f be a topologically transitive nondegenerate almost Anosov diffeomorphism on M with S = {p}. Assume Df_{p} = id, and W_{ϵ}^{u}(p) and W_{ϵ}^{s}(p) are C^{4} curves. With the notation above,

(I) if αa_{2} > 2b_{2} for some 0 < α < 1, a_{1} = 0 = b_{1} and a_{0}b_{2} − a_{2}b_{0} > 0, then f

admits an SBR measure; and (II) if 2a_{2} < αb_{2} for some 0 < α < 1 and a_{1}b_{1} ≠ 0, then f admits an infinite

SBR measure.

The systems satisfying the conditions in Theorem B exist. For example, we can take a torus T^{2} and constants 0 < r_{0} << r_{1} << 1 = diam T^{2} and then construct a dynamical system in such a way that f is a hyperbolic toral automorphism outside the r_{1}−neighborhood and has the form (1.3) and (1.4) within the r_{0}−neighborhood of the origin O.

Remark 1.6. The nondegeneracy conditions (1.1) guarantee that a_{0}, a_{2}, b_{0}, b_{2} > 0. Hence, the conditions in Theorem B (I) imply that ϕ > ψ in some small neighborhood of p, while the conditions in Theorem B (II) imply that in some quadrants, ϕ < ψ near y-axis.

Recall the cases discussed in §0 that Df_{p} ≠ id but one of the eigenvalues is equal to 1. It seems that in such systems the existence of SBR measures depends on whether expansion is “stronger” than contraction. The results in Theorem B is consistent with this observation, because we can think that expansion is “stronger” if ϕ > ψ and is “weaker” if ϕ < ψ.

PART 1: Proof of Theorem A

In this part we always assume that B(p, r_{0}) is in the Euclidean plane IR^{2}, where r_{0} is as in Definition 2. Take a coordinate system in the plane such that the origin is the fixed point p. Thus, we can write |x| = d(x, p). Also, ∀x ∈ B(p, r_{0}), we identify T_{x}M with the same Euclidean plane. Let Θ(x, y) denote the angle from x to y counterclockwise in IR^{2}.

The Assumption A is not going to be used in this part.

2. Dynamics near the Fixed Point

We first prove in Proposition 2.1 that the second derivative D^{2}f_{p} is 0 and the third derivative D^{3}f_{p} is not 0. Then we show that the action of f^{n^{2}} at \frac{x}{n} is similar to the action of f at x, on a smaller scale. This can be seen in Proposition 2.6 and Proposition 2.8, where the former deals with norms and the latter deals with angles.

Proposition 2.1. D^{2}f_{p} = 0 and D^{3}f_{p}(a, a, ·) ≠ 0 ∀a ∈ IR^{2}. So,

Df_{x} = id +\frac{1}{2}D^{3}f_{p}(x, x, ·) + R_{F} (x),

and

fx = x + \frac{1}{6}D^{3}f_{p}(x, x, x) + R_{f}(x),

where R_{F} (x) and R_{f}(x) are remainders with ∥R_{F} (x)∥ = O(|x|^{3}) and |R_{f}(x)| = O(|x|^{4}).

Proof. Since Df_{p} = id, by Taylor expansion

Df_{x} = id +D^{2}f_{p}(x, ·) + \frac{1}{2}D^{3}f_{p}(x, x, ·) + R_{F} (x).  (2.1)

Let ⟨·, ·⟩ denote the inner product. We have

|Df_{x}v|^{2} = |v|^{2} + 2⟨D^{2}f_{p}(x, v), v⟩ + O(|x|^{2})|v|^{2}.

By (1.1) we know that for any sufficiently small x and v ∈ C_{x}^{s}, ⟨D^{2}f_{p}(x, v), v⟩ ≤ 0.

Note that x → C_{x}^{s} is continuous. There exists a small 0 < r < r_{0} such that ⋂_{x∈B(p,r)} C_{x}^{s} has nonempty interior. Thus, we get ⟨D^{2}f_{p}(x, v), v⟩ ≤ 0 for

all x ∈ B(p, r) and v ∈ ⋂_{x∈B(p,r)} C_{x}^{s}. Since ⟨D^{2}f_{p}(·, ·), ·⟩ is trilinear, this implies

⟨D^{2}f_{p}(x, v), v⟩ = 0 ∀x, v ∈ IR^{2}. Now it is easy to conclude that D^{2}f_{p} = 0. The inequality D^{3}f_{p}(a, a, ·) ≠ 0 follows from (1.1), (2.1) and the fact D^{2}f_{p} = 0.

Remark 2.2. Similarly, D^{2}f_{p}^{−1} = 0 and D^{3}f_{p}^{−1}(x, x, ·) ≠ 0 ∀x ∈ IR^{2}.

Next two corollaries follow directly from above facts.

Corollary 2.3. There exists a constant η > 0 such that ∀x ∈ B(p, r_{0}),

∥Df_{x}^{±1} − id ∥ ≤ 3η|x|^{2}  and  |f^{±1}x − x| ≤ η|x|^{3}.

Corollary 2.4. For all x, y ∈ B(p, r_{0}) with d(x, y) ≤ |x|^{2},

d(f^{±1}x, f^{±1}y) ≤ (1 + 4η|f^{±1}x|^{2})d(x, y).

8 HUYI HU

Lemma 2.5. Let x ∈ B(p, r_{0}). If |x| ≥ (3ηk)^{− \frac{1}{2}} for some k > 1, then |fx| ≥ (3η(k + 1))^{− \frac{1}{2}} . Proof. By Corollary 2.3,

|fx| ≥ |x|(1 − η|x|^{2}) ≥ \sqrt{\frac{1}{3kη}} (1 − η · \frac{1}{3kη} ) ≥ \sqrt{\frac{1}{3(k + 1)η}} .

Proposition 2.6. For any ϵ > 0, there exists 0 < r_{ρ} = r_{ρ}(ϵ) ≤ r_{0} such that

∀x ∈ B(p, r_{ρ}), t ∈ (0, 1], j = 1, ⋯, [\frac{2}{t^{2}} ],

(1 − ϵ)|tx| ≤ |f^{j}(tx)| ≤ (1 + ϵ)|tx|.

Proof. Take 0 < r_{ρ} ≤ r_{0} such that e^{2η(1+ϵ)^{2}r_{ρ}^{2}} ≤ (1+ϵ). Let x ∈ B(p, r_{ρ}) arbitrarily. Suppose ∀i = 1, ⋯, j − 1, |f^{i}(tx)| ≤ (1 + ϵ)|tx|. By Corollary 2.3,

\begin{aligned}|f^{j}(tx)| & ≤ |f^{j−1}(tx)|(1 + η|f^{j−1}(tx)|^{2}) ≤ t|x|(1 + η(1 + ϵ)^{2}t^{2}|x|^{2})^{j} \\ & ≤ t|x|(1 + η(1 + ϵ)^{2}t^{2}|x|^{2})^{\frac{2}{t^{2}}} ≤ t|x|e^{2η(1+ϵ)^{2}|x|^{2}} ≤ (1 + ϵ)t|x|. \end{aligned}

Therefore, the second inequality in the lemma follows from induction.

The first inequality can be obtained similarly.

For any 0 ≠ x ∈ B(p, r_{0}), we denote e_{x} = x/|x| or, equivalently, x = |x|e_{x}. Lemma 2.7. For any e_{x} in the unit circle, uniformly

lim_{|t|→0} \frac{Θ(te_{x}, f(te_{x}))}{|t|^{2}} = \frac{1}{6}|D^{3}f_{p}(e_{x}, e_{x}, e_{x})| sin Θ(e_{x}, D^{3}f_{p}(e_{x}, e_{x}, e_{x})).

Proof. Denote A(x) = \frac{1}{6}D^{3}f_{p}(x, x, x). The result follows from the facts |R_{f}(x)| = o(|x|^{3}) as x → 0, A(x) = |x|^{3}A(e_{x}) and

tan Θ(x, fx) = \frac{|A(x)| sin Θ(x, A(x)) + |R_{f}(x)| sin Θ(x, R_{f}(x))}{|x| + |A(x)| cos Θ(x, A(x)) + |R_{f}(x)| cos Θ(x, R_{f}(x))}.

Proposition 2.8. For any ϵ > 0 there exists a constant 0 < r_{θ} = r_{θ}(ϵ) ≤ r_{ρ} such that ∀r ∈ (0, r_{θ}), if x, y ∈ B(p, r) with |Θ(x, y)| ≤ |Θ(x, fx)| and |y| = t|x|, t ∈ (0, 1], then

|Θ(y, f^{j}y)| ≤ |Θ(x, fx)| + ϵ|x|^{2}  ∀ 0 ≤ j ≤ \frac{1}{t^{2}} ,  (2.2)
|Θ(y, f^{j}y)| ≥ |Θ(x, fx)| − ϵ|x|^{2}  ∀ \frac{1}{t^{2}} ≤ j ≤ \frac{2}{t^{2}} .  (2.3)

Moreover, if D^{3}f_{p}(e_{x}, e_{x}, e_{x}) and e_{x} are not collinear, then we can choose r̄_{θ} = r̄_{θ}(ϵ, e_{x}) in such a way that ∀x ∈ B(p, r̄_{θ}),

|Θ(y, f^{j}y)| ≥ |Θ(x, fx)|  ∀ \frac{1}{(1 − ϵ)t^{2}} ≤ j ≤ \frac{2}{t^{2}} .  (2.4)

Proof. Denote A(z) = \frac{1}{6}D^{3}f_{p}(z, z, z).

Note that A(e_{x}) sin Θ(e_{x}, A(e_{x})) is continuous on the unit circle. By Lemma 2.7, there exist r′_{θ} > 0 and θ_{0} > 0 such that ∀x, z ∈ B(p, r′_{θ}) with Θ(x, z) ≤ θ_{0},

\frac{|z|^{2}}{|x|^{2}} |Θ(x, fx)| − \frac{ϵ}{4}|z|^{2} ≤ |Θ(z, fz)| ≤ \frac{|z|^{2}}{|x|^{2}} |Θ(x, fx)| + \frac{ϵ}{4}|z|^{2}.  (2.5)

Also, there exists β ∈ (0, \frac{1}{2}] such that ∀x ∈ B(p, r′_{θ}),

|Θ(x, fx)| ≤ \frac{ϵ}{3β} |x|^{2}.  (2.6)

By Proposition 2.6, there exists r′′_{θ} > 0 such that ∀x ∈ B(p, r′′_{θ}), t ∈ (0, 1],

(1 − β)|tz|^{2} ≤ |f^{j}(tz)|^{2} ≤ (1 + β)|tz|^{2}  j = 0, 1, ⋯, [\frac{2}{t^{2}} ].  (2.7)

Take

r_{θ} = min\{r′_{θ}, r′′_{θ}, \frac{β}{2ϵ}θ_{0}\}.  (2.8)

Now let x, y ∈ B(p, r_{θ}) with |Θ(x, y)| < |Θ(x, fx)| and |y| = t|x|, t ∈ (0, 1].

Suppose for i = 0, 1, ⋯, j, |Θ(x, f^{i}y)| < θ_{0}, where 0 ≤ j ≤ \frac{2}{t^{2}} . By (2.5), (2.7) and (2.6),

|Θ(f^{j}y, f^{j+1}y)| ≤\frac{|f^{j}y|^{2}}{|x|^{2}} (|Θ(x, fx)| + \frac{ϵ}{4}|x|^{2})
≤(1 + β)t^{2}(|Θ(x, fx)| + \frac{ϵ}{4}|x|^{2}) ≤ t^{2}|Θ(x, fx)| + ϵt^{2}|x|^{2}. (2.9)

Similarly,

|Θ(f^{j}y, f^{j+1}y)| ≥ t^{2}|Θ(x, fx)| − ϵt^{2}|x|^{2}.  (2.10)

By (2.9),

|Θ(x, f^{j+1}y)| ≤ |Θ(x, y)| + |Θ(y, f^{j+1}y)| ≤|Θ(x, y)| + 2|Θ(x, fx)| + 2ϵ|x|^{2}
≤ 3|Θ(x, fx)| + 2ϵ|x|^{2}.

Using (2.6) and (2.8) we get

|Θ(x, f^{j+1}y)| ≤ (\frac{ϵ}{β} + 2ϵ) · \frac{β}{2ϵ}θ_{0} ≤ θ_{0}.

Therefore (2.2) and (2.3) can be obtained by induction.

If D^{3}f_{p}(e_{x}, e_{x}, e_{x}) and e_{x} are not collinear, then A(e_{x}) sin Θ(e_{x}, A(e_{x})) ≠ 0. We can use ϵ′ = \frac{ϵ}{2}|A(e_{x})|| sin Θ(e_{x}, A(e_{x}))| instead of ϵ, and therefore obtain r̄_{θ} = r̄_{θ}(ϵ′, e_{x}) > 0 such that if x, y ∈ B(p, r̄_{θ}) with |Θ(x, y)| ≤ |Θ(x, fx)| and |y| = t|x|

for some t ∈ (0, 1], then ϵ′ ≤ ϵ\frac{|Θ(x, fx)|}{|x|^{2}} and (2.10) holds for this ϵ′. That is,

|Θ(f^{j−1}y, f^{j}y)| ≥ t^{2}|Θ(x, fx)| − ϵ′t^{2}|x|^{2} ≥ (1 − ϵ)t^{2}|Θ(x, fx)| j = 0, 1, ⋯, [\frac{1}{t^{2}} ].

Thus (2.4) follows.

3. Eigenvectors of D^{3}f_{p}(x, x, ·) near the Fixed Point

We prove in Proposition 3.2 that id +\frac{1}{2}D^{3}f_{p}(x, x, ·), the approximation of Df_{x} up to the second order of x, has two eigenspaces Ξ_{x}^{u} and Ξ_{x}^{s}. This fact will be used to construct coordinate systems in §5. Usually, the angle between x and Ξ_{x}^{u} is not zero. However, there exists a unique line E^{+} on which every point x and its corresponding Ξ_{x}^{u} are collinear (Lemma 3.6). We will see in §4 that E^{+} is in fact the tangent line, denoted by E_{p}^{u}, of “weak” unstable manifold W^{u}(p) at p.

Lemma 3.1. There exist constants 0 < r̃ ≤ r_{0}, 0 < κ̃^{u} ≤ κ^{u} and 0 < κ̃^{s} ≤ κ^{s}, cones C̃^{u} and C̃^{s} such that ∀x ∈ B(p, r̃), C̃^{u} ⊇ C_{x}^{u}, C̃^{s} ⊇ C_{x}^{s} and

|Df_{x}v| ≥ (1 + κ̃^{u}|x|^{2})|v| ∀v ∈ C̃_{x}^{u},
|Df_{x}v| ≤ (1 − κ̃^{s}|x|^{2})|v| ∀v ∈ C̃_{x}^{u}.

Proof. Denote A_{x} = \frac{1}{2}D^{3}f_{p}(x, x, ·). Also, denote

C_{x}(β) = {v ∈ IR^{2} : ⟨v, A_{x}v⟩ ≥ β|x|^{2}|v|^{2}}.

By Proposition 2.1, Df_{x}v = v + A_{x}v + R_{F} (x)v. Hence, |Df_{x}v|^{2} = |v|^{2} + 2⟨A_{x}v, v⟩ + o(|x|^{2})|v|^{2}. So if |x| is small enough, then

C_{x}^{u} ⊆ C_{x}(\frac{4}{5}κ^{u}) ⊂ C_{x}(\frac{3}{5}κ^{u}) ⊆ \{v ∈ T_{x}M : |Df_{x}v| ≥ (1 + \frac{1}{2}κ^{u}|x|^{2})|v|\}.  (3.1)

Note that C_{tx}^{u}(β) is independent of t provided t ≠ 0. By the continuity of C_{x}^{u}, we get that for any e_{x} in the unit circle, C_{p}^{u} ⊆ C_{e_{x}}(\frac{4}{5}κ^{u}) and therefore C_{p}^{u} ⊆ ⋂_{e_{x}∈S^{1}} C_{e_{x}}(\frac{4}{5}κ^{u}).

Put C̃^{u} = ⋂_{e_{x}∈S^{1}} C_{e_{x}}(\frac{3}{5}κ^{u}). It is easy to see by (3.1) that C̃^{u} is strictly larger than C_{p}^{u}.

Again, by the continuity of C_{x}^{u}, there exists r̃ > 0 such that B(p, r̃) ⊆ {x : C_{x}^{u} ⊆ C̃^{u}}. Now for any x ∈ B(p, r̃) and v ∈ C̃^{u}, we have v ∈ C̃_{x}(\frac{3}{5}κ^{u}). Therefore by (3.1) |Df_{x}v| ≥ (1 + \frac{1}{2}κ^{u}|x|^{2})|v|. This finishes the proof.

Proposition 3.2. For any a ∈ IR^{2}\{0}, id +A_{a} has an eigenvector in C̃^{u}, where A_{a} denotes \frac{1}{2}D^{3}f_{p}(a, a, ·).

Proof. Suppose there is a ∈ IR^{2} with |a| = 1 such that id +A_{a} has no eigenvector in C̃^{u}. We may assume Θ(v, v + A_{a}v) > 0 ∀v ∈ C̃^{u}. Thus there exists θ_{0} > 0 suth that for all unit vector v ∈ C̃^{u}, and unit vectors a′ and v′ with |Θ(a, a′)| ≤ θ_{0} and |Θ(v, v′)| ≤ θ_{0},

|A_{a′}v′| sin Θ(v′, A_{a′}v′) > \frac{1}{2}|A_{a}v| sin Θ(v, A_{a}v).  (3.2)

Take x = ta for some t ∈ (0, \frac{1}{2}) such that Θ(x, fx) + \frac{1}{2}|x|^{2} ≤ θ_{0}.

Let v̄_{z} be a unit vector in the boundary of C_{z}^{u} satisfying Θ(v, v̄_{z}) ≥ 0 ∀v ∈ C_{z}^{u}. By continuity we can take n > 0 such that for any z_{1}, z_{2} ∈ B(p, \frac{1}{n}),

Θ(v̄_{z_{1}}, v̄_{z_{1}}) < min_{v∈C̃^{u}}\{θ_{0}, \frac{1}{32}|A_{x}v| sin Θ(v, A_{x}v)\}.  (3.3)

We may assume that n is large enough such that

Θ(v, Df_{z}v) > \frac{1}{2}|A_{z}v| sin Θ(v, A_{z}v)  ∀v ∈ C̃^{u}.  (3.4)

provided z ∈ B(p, \frac{1}{n}) with |Θ(x, z)| ≤ θ_{0}. This is possible because

tan Θ(v, Df_{z}v) = \frac{|A_{z}v| sin Θ(v, A_{z}v) + |R_{F} (z)| sin Θ(v, R_{F} (z)v)}{|v| + |A_{z}v| cos Θ(v, A_{z}v) + |R_{F} (z)| cos Θ(v, R_{F} (z)v)} .

Put y = \frac{x}{n}. Denote y_{i} = f^{i}y and v̄^{(i)} = \frac{Df_{y}^{i}v̄_{y}}{|Df_{y}^{i}v̄_{y}|}. We may also assume that

Proposition 2.6 and Proposition 2.8 can be applied with ϵ = \frac{1}{2}. Therefore Θ(x, y_{i}) ≤

Θ(x, fx) + \frac{1}{2}|x|^{2} ≤ θ_{0} and \frac{|x|}{2n} ≤ |y_{i}| ≤ \frac{3|x|}{2n} ≤ \frac{1}{n} ∀i = 0, 1, ⋯, n^{2}.

Note that Df_{y}^{i}C_{y}^{u} ⊆ C_{y_{i}}^{u}. We have that ∀i = 0, 1, ⋯, n^{2}, v̄^{(i)} ∈ C_{y_{i}}^{u} ⊆ C̃^{u} and

Θ(v̄_{y_{0}}, v̄_{y_{i}}) = Θ(v̄_{y_{0}}, Df_{x}^{i}v̄_{y_{0}}) + Θ(v̄^{(i)}, v̄_{y_{i}}) ≥ Θ(v̄_{y_{0}}, Df_{x}^{i}v̄_{y_{0}}).  (3.5)

By (3.4) and (3.2) we get

Θ(v̄_{y_{0}}, Df_{x}^{n^{2}} v̄_{y_{0}}) = ∑_{i=0}^{n^{2}−1} Θ(v̄^{(i)}, Df_{y_{i}}v̄^{(i)}) ≥ ∑_{i=0}^{n^{2}−1} \frac{1}{2}|A_{y_{i}}v̄^{(i)}| sin Θ(v̄^{(i)}, A_{y_{i}}v̄^{(i)})

≥n^{2} · \frac{1}{2} · |\frac{x}{2n}|^{2} · \frac{1}{2}|A_{e_{x}} v̄_{x}| sin Θ(v̄_{x}, A_{x}v̄_{x}) ≥ \frac{1}{16}|A_{x}v̄_{x}| sin Θ(v̄_{x}, A_{x}v̄_{x}),

It contradicts to (3.5) and (3.3).

We denote by ξ_{a}^{u} and ξ_{a}^{s} the unit eigenvectors of id +A_{a} in C̃^{u} and C̃^{s} respectively, and by 1 + λ_{a}^{u}|a|^{a} and 1 − λ_{a}^{s}|a|^{2} the corresponding eigenvalues. Also, denote by Ξ_{a}^{u} and Ξ_{a}^{s} the subspaces generated by ξ_{a}^{u} and ξ_{a}^{s}. Since A_{ta} = t^{2}A_{a}, we know

ξ_{ta}^{u} = ξ_{a}^{u}, ξ_{ta}^{s} = ξ_{a}^{s}, and λ_{ta}^{u} = λ_{a}^{u}, λ_{ta}^{u} = λ_{a}^{u}.  (3.6)

By Lemma 3.1 and Corollary 2.3, κ̃^{u} ≤ λ_{a}^{u} ≤ 3η and κ̃^{s} ≤ λ_{a}^{s} ≤ 3η.

Lemma 3.3. For any a_{1}, ⋯, a_{n} ∈ IR^{2}\{0}, ∏_{i=1}^{n}(id +A_{a_{i}}) has an eigenvector in

C̃^{u}, where A_{a} = \frac{1}{2}D^{3}f_{p}(a, a, ·).

Proof. By the proof of Proposition 3.2, we have ξ_{x}^{u} ∈ int C̃^{u}. (Otherwise we can shrink C̃^{u} a little bit and the arguments still work.) This implies (id +A_{x})C̃^{u} ⊂ C̃^{u}.

So ∏_{i=0}^{n}(id +A_{a_{i}})C̃^{u} ⊂ C̃^{u}. Then the result follows.

Lemma 3.4. If r is small enough,then ∀x∈ B(p, r), Df_{x}C̃^{u} ⊂ C̃^{u} and Df_{x}C̃^{s} ⊃ C̃^{s}.

Proof. This is because (id +A_{x})C̃^{u} ⊂ C̃^{u} and ∥Df_{x} − (id +A_{x})∥ = O(|x|^{3}).

Remark 3.5. Df_{x} has an eigenvector in C̃^{u} and an eigenvector in C̃^{s} if |x| is small.

Lemma 3.6. For a, b ∈ IR^{2}\{0}, denote the matrix of identity Ξ_{a}^{u} ⊕ Ξ_{a}^{s} → Ξ_{b}^{u} ⊕ Ξ_{b}^{s} by

( \begin{matrix}1 + h_{11}(a, b) & h_{12}(a, b) \\ h_{21}(a, b) & 1 + h_{22}(a, b) \end{matrix} ) .

Then ∀i, j = 1, 2, |h_{ij}(a′, b) − h_{ij}(a′′, b)| ≤ C|Θ(a′, a′′)|, |h_{ij}(a, b′) − h_{ij}(a, b′′)| ≤ C|Θ(b′, b′′)|, and |h_{ij}(a, b)| ≤ C|Θ(a, b)|, where C is a generic constant depending only on f.

Proof. Note that ∀a ≠ 0, A_{a} has one positive eigenvalue and one negative eigenvalue. So it is clear that the maps a → ξ_{a}^{u}, ξ_{a}^{u} are smooth on IR^{2}\{0}. By (3.6) the result follows.

Lemma 3.7. There exists unique subspace E^{+} ⊆ IR^{2} such that ∀a ∈ E^{+}\{0}, a is an eigenvector of A_{a} with positive eigenvalue, where A_{a} = \frac{1}{2}D^{3}f_{p}(a, a, ·)

Proof. The existence follows from the continuity of the map a → Θ(a, A_{a}a).

Now we suppose that there exist x, y with x, y ∈ C̃^{u} and Θ(x, y) > 0 such that A_{x}x = λ_{x}^{u}|x|^{2}x and A_{y}y = λ_{y}^{u}|y|^{2}y.

First, we assume λ_{x}^{u} = λ_{y}^{u}. Take such x and y with x − y ∈ C̃^{s} and |x| ≠ |y|. By

the law of cosines, Lemma 2.7 and the fact that |f(tx)| = t|x|+ \frac{1}{3}λ_{x}^{u}t^{3}|x|^{3} +O(|t|^{4}), we obtain that

\begin{matrix}\frac{3}{t^{4}} (|f(ty) − f(tx)|^{2} − |ty − tx|^{2}) \\ = & 2λ_{x}^{u}|x|^{4} + 2λ_{y}^{u}|y|^{4} − 2|x||y|(λ_{x}^{u}|x|^{2} + λ̄_{y}^{u}|y|^{2}) cos Θ(y, x) + O(t) \end{matrix}

as t → 0^{+}. Since

λ_{x}^{u}|x|^{4} + λ_{y}^{u}|y|^{4} − |x||y|(λ_{x}^{u}|x|^{2} + λ_{y}^{u}|y|^{2}) = (λ_{x}^{u}|x|^{3} − λ_{y}^{u}|y|^{3})(|x| − |y|),

The right hand side is positive if t is small. So |f(ty) − f(tx)| > |ty − tx|. But the fact y − x ∈ C̃^{s} implies |f(ty) − f(tx)| < |ty − tx|, a contradiction.

Next, we assume λ_{x}^{u} ≠ λ_{y}^{u}. In this case, for any x and y, f(tx) − f(ty) is not parallel to x − y. Hence we can find x and y such that x − y ∉ C̃^{s} but f(tx) − f(ty) ∈ C̃^{s}. Therefore, there is a piont tz between tx and ty such that Df_{tz} maps a vector outside the cone C̃^{s} into C̃^{s}. This contradicts Lemma 3.4 that says Df_{tz}C̃^{s} ⊃ C̃^{s} if t is small.

4. Unstable Manifolds on M

In this section we prove the existence of invariant decomposition of the tangent bundle into T M = E^{u} ⊕ E^{s} (Proposition 4.2) and the existence of “weak” unstable manifolds W^{u}(x) := {y ∈ M : lim_{n→∞} d(f^{−n}x, f^{−n}y) = 0} (Proposition 4.4). E_{x}^{u} is tangent to W^{u}(x) for all x ∈ M and is continuous everywhere else except the fixed point p. Most of the arguments are routine except for proving that E_{p}^{u} is tangent to W^{u}(p) (Lemma 4.5).

For convenience we will refer to W^{u}(x) and W_{ϵ}^{u}(x) = {y ∈ M : d(f^{−n}x, f^{−n}y) ≤ ϵ ∀n ≥ 0} as the “unstable manifold” and “local unstable manifold” at x, even though points on the manifolds may not be contracted exponentially in backwards time.

Lemma 4.1. Let x ∈ M\{p}. Then |Df_{x}^{n}v| → ∞ ∀v ∈ C_{x}^{u} and |Df_{x}^{−n}v| → ∞ ∀v ∈ C_{x}^{s}, as n → ∞.

Proof. If f^{n}x ∉ B(p, r_{0}) for infinite number of n’s, then by (1.2) the result is clear. If there exists N > 0 such that ∀n > N, f^{n}x ∈ B(p, r_{0}), then by Lemma 2.5,

|Df_{x}^{n}v| ≥ |Df_{x}^{N}v| ∏_{i=N+1}^{n−1}(1 + κ^{u}|f^{i}x|^{2}) ≥ |Df_{x}^{N}v| ∏_{i=N+1}^{n−1}(1 + κ^{u} \frac{1}{3(k + i)η} )

for some k > 1. Thus |Df_{x}^{n}v| → ∞.

Proposition 4.2. There exists an invariant decomposition of the tangent bundle into T M = E^{u} ⊕ E^{s} such that ∀x ∈ M, E_{x}^{u} ⊂ C_{x}^{u}, E_{x}^{s} ⊂ C_{x}^{s}, and Df_{x}E_{x}^{u} = E_{fx}^{u}, Df_{x}E_{x}^{s} = E_{fx}^{s}. Except for the fixed point p, the decomposition is continuous.

Proof. E_{p}^{u} and E_{p}^{s} have been defined in the above section.

For x ∈ M\{p}, let E_{x}^{u} = ⋂_{n=0}^{∞} Df_{f^{−n}x}^{n}C_{f^{−n}x}^{u}. Clearly, E_{x}^{u} ⊂ C_{x}^{u} and Df_{x}E_{x}^{u} = E_{fx}^{u}.

We show that E_{x}^{u} is one dimensional subspace in T_{x}M. In fact, if there are two independent vectors in E_{x}^{u}, then we can choose v, v′ ∈ E_{x}^{u} such that 0 ≠ v − v′ ∈ C_{x}^{s}. By Lemma 4.1 |Df_{x}^{−n}(v − v′)| → ∞ as n → ∞. On the other hand, since v, v′ ∈ Df_{f^{−n}x}^{n}C_{f^{−n}x}^{u}, |Df_{x}^{−n}(v − v′)| ≤ |Df_{x}^{−n}v′| + |Df_{x}^{−n}v| ≤ |v| + |v′| ∀n > 0. This is a contradiction.

Put E_{x}^{u} = E_{x}^{u} ∀x ∈ M\{p}.

Now we prove the continuity of E_{x}^{u}. Suppose for some x_{0} ∈ M\{p}, there exists a sequence {x_{i}} such that lim_{i→∞} x_{i} = x_{0} and lim_{i→∞} E_{x_{i}}^{u} = E′_{x_{0}} ≠ E_{x_{0}}^{u}. Take v ∈ E′_{x_{0}}

with |v| = 1. Let v^{u} and v^{s} denote the projection of v in E_{x_{0}}^{u} and E_{x_{0}}^{s} respectively.

Since E′_{x_{0}} ≠ E_{x_{0}}^{u}, v^{s} ≠ 0. By Lemma 4.1 |Df_{x_{0}}^{−n}v^{s}| → ∞ as n → ∞. Hence, we can find an n > 0 such that |Df_{x_{0}}^{−n}v| ≥ 2. On the other hand, if we take v_{i} ∈ E_{x_{i}}^{u} ∀i > 0 such that lim_{i→∞} v_{i} = v, then by the continuity of Df_{x}^{−n}, we have

|Df_{x_{0}}^{−n}v| = lim_{i→∞} |Df_{x_{i}}^{−n}v_{i}| ≤ 1, a contradiction.

Remark 4.3. The decomposition into T_{x}M = E_{x}^{u} ⊕ E_{x}^{s} is not continuous at p. This can be seen by using a similar method as in the proof of Proposition 3.2 for vectors in unstable subspaces instead of those in the boundaries of unstable cones. We leave details to the reader.

Proposition 4.4. For any x ∈ M, W_{ϵ}^{u}(x) is a curve tangent to E^{u}.

Proof. Let x ∈ M\{p}. Construct a continuous vector field v_{y} in a suitable neighborhood of x such that v_{y} ∈ E_{y}^{u}. It is easy to see that the integral curve of the vector field that passes throuth x is contained in to W^{u}(x). To prove that W^{u}(x) is also contained in the integral curve passing throuth x, it is enough to show that any piece of stable curve γ^{s} intersects W_{ϵ}^{u}(x) at most one point.

In fact, if γ^{s} ∩ W_{ϵ}^{u}(x) ⊃ {y, z} with y ≠ z, then by Lemma 4.1, we have

lim_{n→∞} d(f^{−n}y, f^{−n}x) = 0 and lim_{n→∞} d(f^{−n}z, f^{−n}x) = 0. So lim_{n→∞} d(f^{−n}y, f^{−n}z) = 0.

This contradicts to the fact that Df^{−n} is expanding along the tangent lines of γ^{s}.

Now we consider the case x = p. Let Ω be the set of points in B(p, ϵ) that can be jointed from p by a curve tangent to vectors in C̃^{u}. Note that f^{n}(Ω) ∩ B(p, ϵ)

is decreasing as n → ∞. It is easy to check W_{ϵ}^{u}(p) = ⋂_{i=0}^{∞}(f^{i}Ω ∩ B(p, ϵ)). Above

argument on uniqueness shows that W_{ϵ}^{u}(p) is a curve. The differentiability of W_{ϵ}^{u}(p) at p is proved in Lemma 4.5.

Lemma 4.5. W_{ϵ}^{u}(p) is tangent to E_{p}^{u}.

Proof. Let γ : [−1, 1] → W_{ϵ}^{u}(p) be the parameter expression with γ(0) = p. For x_{1}, x_{2} ∈ W_{ϵ}^{u}(p), we say x_{1} > x_{2}, if x_{1} = γ(s_{1}), x_{2} = γ(s_{2}) and s_{1} > s_{2}.

First, we prove that the one side limit lim_{s→0^{+}} \frac{γ(s)}{|γ(s)|} exists. Suppose there are two

sequences x′_{n} = γ(s′_{n}), x′′_{n} = γ(s′′_{n}), s′_{n}, s′′_{n} → 0 as n → ∞, and two unit vectors e′,

e′′ such that \frac{x′_{n}}{|x′_{n}|} = e′ and \frac{x′′_{n}}{|x′′_{n}|} = e′′ ∀n > 0. Without loss generality we assume

Θ(e′, e′′) ≥ 0 and x′_{1} > x′′_{1} > x′_{2} > x′′_{2} > · · ·. Let e = \frac{β′e′ + β′′e′′}{|β′e′ + β′′e′′|} for some

β′, β′′ > 0. Take y_{n}, z_{n} ∈ W_{ϵ}^{u}(p) such that

y_{n} = sup{y < x′_{n} : y = |y|e}  and  z_{n} = sup{z < x′′_{n} : z = |z|e}.

By Corollary 2.3, |fy_{n} − y_{n}| ≤ η|y_{n}|^{3}. Thus, if n is large enough, then y_{n} < fy_{n} < x′_{n}, and therefore Θ(fy_{n}, y_{n}) and Θ(x′_{n}, y_{n}) have same sign, i.e. Θ(fy_{n}, y_{n}) > 0. Similarly, we have Θ(fz_{n}, z_{n}) < 0. Thus, by Lemma 2.7 we get Θ(e, D^{3}f_{p}(e, e, e)) = 0. Since β′ and β′′ are arbitrary, by Lemma 3.7 we must have e′ = e′′. So the limit

e_{+}^{u} := lim_{s→0^{+}} \frac{γ(s)}{|γ(s)|} exists.

Now we prove e_{+}^{u} ∈ E_{p}^{u}. Suppose it is not true. We may assume Θ(e_{+}^{u}, A_{e_{+}^{u}}e_{+}^{u}) >

0. By Lemma 2.7 lim_{t→0^{+}} \frac{Θ(te_{+}^{u}, f(te_{+}^{u}))}{t^{2}} > 2β for some β > 0. Hence, we can

find t_{0} > 0, θ_{0} > 0 such that for any z ∈ Γ := {y : |y| ≤ t_{0}, |Θ(y, e_{+}^{u})| ≤ θ_{0}}, Θ(z, fz) > β|z|^{2}. Also, we can find s_{0} > 0 such that the piece of unstable curve {γ(s) : 0 ≤ s ≤ s_{0}} is contained in Γ. Take any point x in this curve. We have

Θ(f^{−n}x, x) = ∑_{i=1}^{n} Θ(f^{−i}x, f^{−i+1}x) ≥ β ∑_{i=1}^{n} |f^{−i}x|^{2}. By Lemma 2.5, ∑_{i=1}^{n} |f^{−i}x|^{2} is

unbounded. This contradicts the fact that f^{−i}x ∈ Γ ∀i ≥ 0.

Similarly, the limit e_{−}^{u} := lim_{s→0^{−}} \frac{γ(s)}{|γ(s)|} exists and satisfies e_{−}^{u} ∈ E_{p}^{u}.

Though E_{x}^{u} and E_{x}^{s} are not continuous at p, they are contained in C_{x}^{u} and C_{x}^{s}. Therefore we know that f has a local product structure, i.e, there exist constants ϵ > 0, δ > 0, such that ∀x, y ∈ M with d(x, y) ≤ δ, [x, y] := W_{ϵ}^{u}(x) ∩ W_{ϵ}^{u}(y) contains exact one point.

Lemma 4.6. Let x ∈ B(p, r_{0}) and y be in the W^{u}-segment connecting x and [x, fx]. Then

d(x, y) ≤ Cd(x, fx)  and  d(x, y) ≤ ηC|x|^{3},

for some a generic constant, which is allowed to depend only on f.

Proof. Use the fact that C_{x}^{u} and C_{x}^{s} are bounded away from each other to get the first inequality. The second one then follows from Corollary 2.3.

5. Coordinate Systems

The purpose of this section is to choose a suitable coordinate system at each tangent space, under which we can prove that at most points in M, Df_{x} contract angles between vectors in C̃^{u}. This is inportant for the proof of local Höder condition in next section.

Let Ẽ_{x}^{u}⊕Ẽ_{x}^{s} be any coordinate systems such that Ẽ_{x}^{u} ⊂ C_{x}^{u} and Ẽ_{x}^{s} ⊂ C_{x}^{s} ∀x ∈ M. Under the coordinate systems, for each x ∈ M there exists a correspondence π_{x} from the unit circle in the tangent space T_{x}M to IR ∪ {∞} such that if π_{x}e = σ, then e and (1, σ) are collinear. The correspondence is one to one if we identify −e with e.

For simplicity of notations, we say σ_{x} ∈ C_{x}^{u} if π_{x}^{−1}σ_{x} ∈ C_{x}^{u}, and σ ∈ C^{u} if σ_{x} ∈ C_{x}^{u} ∀x ∈ M. In particular, we denote by σ^{u} the unique function that satisfies σ ∈ E^{u}.

Suppose (f_{ij}(x)) is the matrix expression of Df_{x} : Ẽ_{x}^{u} ⊕ Ẽ_{x}^{s} → Ẽ_{fx}^{u} ⊕ Ẽ_{fx}^{s}. With respect to the coordinate systems, Df_{x} induces a map F_{x} on IR given by

F_{x}σ = (f_{21}(x) + f_{22}(x)σ)(f_{11}(x) + f_{12}(x)σ)^{−1}  ∀σ ∈ IR.  (5.1)

For any σ, we define Fσ = F ∘ σ ∘ f^{−1}, i.e. (Fσ)(x) = F_{f^{−1}x}σ(f^{−1}x). Clearly, Fσ ∈ C^{u} if σ ∈ C^{u} and Fσ ∈ E^{u} if σ ∈ E^{u}.

Lemma 5.1. For any σ ∈ C^{u}, lim_{n→∞}(F^{n}σ)(x) = σ^{u}(x) ∀x ∈ M\{p}.

Proof. Note that (F^{n}σ)(x) ∈ Df_{f^{−n}x}^{n}C_{f^{−n}x}^{u}. By the proof of Proposition 4.2,

lim_{n→∞}(F^{n}σ)(x) ∈ ⋂_{n=0}^{∞} Df_{f^{−n}x}^{n}C_{f^{−n}x}^{u} = E_{x}^{u}.

Consider a particular coordinate system Ξ_{x}^{u} ⊕ Ξ_{x}^{s} on T_{x}M ∀x ∈ B(p, r_{0}), where Ξ_{x}^{u} and Ξ_{x}^{s} are eigenspaces of A_{x} = \frac{1}{2}D^{3}f_{p}(x, x, ·).

Lemma 5.2. Suppose r_{0} > 0 is small sufficiently. Then with respect to the decomposition T M_{x} = Ξ_{x}^{u} ⊕ Ξ_{x}^{s} ∀x ∈ B(p, r_{0}), the induced map F_{x} of Df_{x} satisfies the following.

i) There exists a constant C_{0} > 0 such that ∀σ ∈ C̃^{u},

|F_{x}σ − F_{y}σ| ≤ C_{0}|x|^{−1}d(x, y)  ∀x ∈ B(p, r_{0}), y ∈ B(x, |x|^{3}).  (5.2)

ii) There exist constants κ′, η′ > 0, sectors S_{r_{0}}^{u} = {y ∈ B(p, r_{0}) : |Θ(y, E_{p}^{u})| ≤ θ^{u}} and S_{r_{0}}^{s} = {y ∈ B(p, r_{0}) : |Θ(y, E_{p}^{s})| ≤ θ^{s}} for some θ^{u}, θ^{s} > 0 such

that ∀σ_{1}, σ_{2} ∈ C̃_{x}^{u},
|F_{x}(σ_{1} − σ_{2})| ≤ (1 − κ′|x|^{2})|σ_{1} − σ_{2}|  ∀x ∈ S_{r_{0}}^{u} ∪ S_{r_{0}}^{s},  (5.3)
|F_{x}(σ_{1} − σ_{2})| ≤ (1 + η′|x|^{2})|σ_{1} − σ_{2}|  ∀x ∈ B(p, r_{0})\(S_{r_{0}}^{u} ∪ S_{r_{0}}^{s}). (5.4)

Proof. Note that Df_{x}ξ_{x}^{u} = (1 + λ_{x}^{u}|x|^{2})ξ_{x}^{u} + R_{F} (x)ξ_{x}^{u} and Df_{x}ξ_{x}^{s} = (1 − λ_{x}^{s}|x|^{2})ξ_{x}^{s} + R_{F} (x)ξ_{x}^{s}. The matrix of Df_{x} : Ξ_{x}^{u} ⊕ Ξ_{x}^{s} → Ξ_{fx}^{u} ⊕ Ξ_{fx}^{s} can be expressed as

( \begin{matrix}1 + g_{11}(x) & g_{12}(x) \\ g_{21}(x) & 1 + g_{22}(x) \end{matrix} )
=(\begin{matrix}1 + h_{11}(x, fx) & h_{12}(x, fx) \\ h_{21}(x, fx) & 1 + h_{22}(x, fx) \end{matrix})(\begin{matrix}1 + λ_{x}^{u}|x|^{2} + O(|x|^{2}) & O(|x|^{2}) \\ O(|x|^{2}) & 1 − λ_{x}^{s}|x|^{2} + O(|x|^{2}) \end{matrix}),(5.5)

where h_{ij} are as in Lemma 3.6.

By (5.1) we get

F_{x}σ = (g_{21}(x) + σ + g_{22}(x)σ)(1 + g_{11}(x) + g_{12}(x)σ)^{−1},  (5.6)
|F_{x}(σ_{1} − σ_{2})|= \frac{1 + g_{11}(x) + g_{22}(x) + g_{11}(x)g_{22}(x) − g_{12}(x)g_{21}(x)}{(1 + g_{11}(x) + g_{12}(x)σ)^{2}}|σ_{1} − σ_{2}|, (5.7)

where σ in (5.7) is between σ_{1} and σ_{2}.

By Lemma 3.6, |h_{ij}(x, fx)−h_{ij}(y, fy)|≤C|Θ(x, y)|+C|Θ(fx, fy)|≤3C|Θ(x, y)|. It is easy to see by (3.6) and the linearity of A_{x} that |λ_{x}^{u} −λ_{y}^{u}| ≤ 3C|Θ(x, y)|, if C is large enough. Note that sin |Θ(x, y)| ≤ |x|^{−1}d(x, y) and ||x|^{2} − |y|^{2}| ≤ 3|x|d(x, y). Therefore, (5.2) follows from (5.5) and (5.6).

SBR MEASURES FOR “ALMOST ANOSOV” DIFFEOMORPHISMS 17

By Lemma 2.7, |Θ(x, fx)| = O(|x|^{2}). So (5.4) follows from (5.5) and (5.7).

Moreover, if x ∈ E_{p}^{u} or E_{p}^{s}, then Θ(x, fx) = o(|x|^{2}). Hence h_{ij}(x, fx) = o(|x|^{2}) as x → 0. So g_{11}(x) = λ_{x}^{u}+o(|x|^{2}), g_{22}(x) = −λ_{x}^{s}+o(|x|^{2}), and g_{12}(x), g_{21}(x) = o(|x|^{2}). Thus, (5.3) holds for these x. Then we use continuity.

Let β > 0 small. For x ∈ B(p, r_{0}), take k = k(x) such that

1 + 2β ≤ ∏_{i=0}^{k−1} (1 + |f^{i}x|^{2}) ≤ 1 + 4β.

Considering Proposition 2.6, we have that there exist 0 < c_{1} ≤ c_{2} ≤ 1 such that

c_{1} ≤ k|x|^{2} ≤ c_{2}.  (5.8)

Lemma 5.3. There exists C′_{β} > 0 such that

∥Df_{x}^{k} − Df_{y}^{k}∥ ≤ kC′_{β}|x|d(x, y) ∀x ∈ B(p, r_{0}), y ∈ B(x, |x|^{3}). Proof. We have

∥Df_{x}^{k} − Df_{y}^{k}∥ ≤ ∑_{j=0}^{k−1} ∥Df_{f^{j+1}x}^{k−j−1}∥∥Df_{f^{j}x} − Df_{f^{j}y}∥∥Df_{y}^{j}∥.

By Proposition 2.1, ∥Df_{f^{j}x} − Df_{f^{j}y}∥ ≤ C′|f^{j}x|d(f^{j}x, f^{j}y) for some C′ > 0. Also,

by By Corollary 2.3, ∥Df_{x}^{j}∥ ≤ ∏_{i=0}^{j−1}(1 + 3η|f^{i}x|^{2}) ≤ ∏_{i=0}^{j−1}(1 + |f^{i}x|^{2})^{3η}. Since the

difference among x, |f^{i}x| and |f^{i}y| are of higher order, we have

∥Df_{x}^{k} − Df_{y}^{k}∥ ≤ kC′(1 + 4β)^{3η}|x|d(x, y).

Suppose (f_{ij}^{(k)}(x)) is the matrix expression of Df_{x}^{k} : Ξ_{x,k}^{u}⊕Ξ_{x,k}^{s} → Ξ_{f^{k}x,k}^{u}⊕Ξ_{f^{k}x,k}^{s}, where Ξ_{x,k}^{u} and Ξ_{x,k}^{s} are eigenspaces of ∏_{i=0}^{k−1}(id +\frac{1}{2}D^{3}f_{p}(f^{i}x, f^{i}x, ·)). It induces a map on IR by

F_{x}^{(k)}σ = (f_{21}^{(k)}(x) + f_{22}^{(k)}(x)σ)(f_{11}^{(k)}(x) + f_{12}^{(k)}(x)σ)^{−1} ∀σ ∈ IR. (5.9)

Lemma 5.4. The map F_{x}^{(k)} satisfies the following.

i) There exists a constant C_{β} > 0 such that ∀σ ∈ C̃^{u},

|F_{x}^{(k)}σ − F_{y}^{(k)}σ| ≤ C_{β}|x|^{−1}d(x, y) ∀x ∈ B(p, r_{0}), y ∈ B(x, |x|^{3}). (5.10)
ii) There exist constants 1 > C_{κ} > 0, C_{η} > 0 such that ∀σ_{1}, σ_{2} ∈ C̃_{x}^{u},
|F_{x}^{(k)}(σ_{1} − σ_{2})| ≤ C_{κ}|σ_{1} − σ_{2}|  ∀x ∈ S_{r_{0}}^{u} ∪ S_{r_{0}}^{s},  (5.11)
|F_{x}^{(k)}(σ_{1} − σ_{2})| ≤ C_{η}|σ_{1} − σ_{2}|  ∀x ∈ B(p, r_{0})\(S_{r_{0}}^{u} ∪ S_{r_{0}}^{s}).  (5.12)

Proof. The matrix (f_{ij}^{(k)}(x)) can be written as

( \begin{matrix}f_{11}^{(k)}(x) & f_{12}^{(k)}(x) \\ f_{21}^{(k)}(x) & f_{22}^{(k)}(x) \end{matrix} )=(\begin{matrix}1 + h_{11}^{(k)}(x, f^{k}x) & h_{12}^{(k)}(x, f^{k}x) \\ h_{21}^{(k)}(x, f^{k}x) & 1 + h_{22}^{(k)}(x, f^{k}x) \end{matrix})( \begin{matrix}f̄_{11}^{(k)}(x) & f̄_{12}^{(k)}(x) \\ f̄_{21}^{(k)}(x) & f̄_{22}^{(k)}(x) \end{matrix} ) , (5.13)

where (f̄_{ij}^{(k)}(x)) is the matrix of Df_{x}^{k} with respect to the coordinate system Ξ_{x,k}^{u} ⊕ Ξ_{x,k}^{s}, and (δ_{ij} + h_{ij}^{(k)}(x)) is the matrix of identity Ξ_{x,k}^{u} ⊕ Ξ_{x,k}^{s} → Ξ_{f^{k}x,k}^{u} ⊕ Ξ_{f^{k}x,k}^{s}.

By (5.9),

|F_{x}^{(k)}(σ_{1} − σ_{2})|= \frac{f_{11}^{(k)}(x)f_{22}^{(k)}(x) − f_{12}^{(k)}(x)f_{21}^{(k)}(x)}{(f_{11}^{(k)}(x) + f_{12}^{(k)}(x)σ)^{2}}|σ_{1} − σ_{2}|,  (5.14)

where σ is between σ_{1} and σ_{2}.

Similarly, we have |h_{ij}^{(k)}(x, f^{k}x) − h_{ij}^{(k)}(y, f^{k}y)| ≤ C|Θ(x, y)| + C|Θ(f^{k}x, f^{k}y)| ≤ 3C|Θ(x, y)| ≤ 4C|x|^{−1}d(x, y). By (5.8) and Lemma 5.3, |f̄_{21}^{(k)}(x) − f̄_{21}^{(k)}(y)| ≤ C′_{β}k|x|d(x, y) ≤ C′_{β}|x|^{−1}d(x, y). Hence (5.10) follows from (5.13), (5.8) and (5.9).

Since f̄_{ij}^{(k)}(x) and h_{ij}^{(k)}(x, f^{k}x) are bounded, (5.12) follows from (5.13) and (5.14). Now it remains to prove (5.11). Note that ∀v ∈ Ξ_{f^{k}x,k}^{u} with |v| = 1,

|(∏_{i=0}^{k−1}(id +A_{f^{i}x})) v| ≥ ∏_{i=0}^{k−1}(1 + κ^{u}|f^{i}x|) ≥ (1 + β)^{κ^{u}},

where A_{x} = \frac{1}{2}D^{3}f_{p}(x, x, ·). Also note that by Proposition 2.1 and (5.8),

∥Df_{x}^{k} − ∏_{i=0}^{k−1}(id +A_{f^{i}x})∥ = kO(|x|^{3}) = O(|x|).

So given ϵ > 0, we can take r_{0} such that f̄_{11}^{(k)}(x) ≥ (1 + β)^{κ^{u}} − ϵ, and |f̄_{12}^{(k)}(x)| ≤ ϵ ∀x ∈ B(p, r_{0}). For the same reason, we have |f̄_{21}^{(k)}(x)| ≤ ϵ, f̄_{22}^{(k)}(x) ≤ (1 − β)^{κ^{s}} + ϵ.

Since x ∈ S_{r_{0}}^{u} ∪ S_{r_{0}}^{s}, fx, ⋯, f^{k}x are close to E_{p}^{u} or E_{p}^{s}, and the eigenvectors are close to Ξ_{x}^{u} and Ξ_{x}^{s}. Hence |h_{ij}^{(k)}(x, f^{k}x)| ≤ ϵ. Now we obtain that the matrics (f_{ij}^{(k)}(x)) satisfies the folloing condition:

\begin{matrix}f_{11}^{(k)}(x) ≥ (1 + β)^{κ^{u}} − cϵ, & |f_{12}^{(k)}(x)| ≤ cϵ, \\ |f_{21}^{(k)}(x)| ≤ cϵ, & f_{22}^{(k)}(x) ≤ (1 − β)^{κ^{s}} + cϵ, \end{matrix}

where c is a constant independent of ϵ. So we can take ϵ small enough such that (5.14) implies (5.11).

For r ∈ (0, r_{0}], denote S_{r}^{u} = B(p, r) ∩ S_{r_{0}}^{u}, S_{r}^{s} = B(p, r) ∩ S_{r_{0}}^{s} and S_{r}^{c} = B(p, r)\(S_{r}^{u} ∪ S_{r}^{s}).

Let Λ ⊂ M. For x ∈ Λ, denote by τ_{Λ}^{±}(x) the maximal nonnegative integers such that ∀k = 0, 1, ⋯ τ_{Λ}^{±}(x), f^{±k}x ∈ Λ.

Lemma 5.5. Let β ≥ 0. For each x ∈ S_{r_{0}}^{c}, take −τ_{S_{r_{0}}^{c}}^{−} (x) = n_{0} < n_{1} < ⋯ <

n_{m} = τ_{S_{r_{0}}^{c}}^{+} (x) such that 1 + 2β ≤ ∏_{i=n_{j−1}}^{n_{j}−1}(1 + |f^{i}x|^{2}) ≤ 1 + 4β ∀j = 1, ⋯, m. Then

m is bounded for all x ∈ S_{r_{0}}^{c}. Proof. Denote τ^{±}(x) = τ_{S_{r_{0}}^{c}}^{±} (x). Fix an z ∈ S_{r_{0}}^{c} such that f^{−τ^{−}(z)−1}(z) ∈ S_{r_{0}}^{s} and f^{τ^{+}(z)+1}(z) ∈ S_{r_{0}}^{u}. We may assume e^{4|f^{j}z|^{2}} ≤ 1 + 4β ∀ − τ^{−}(z) − 1 ≤ j ≤ τ^{+}(z).

We only need consider the case that |f^{i}x| ≤ \frac{1}{2} min\{|f^{j}z| : −τ^{−}(z) − 1 ≤ j ≤ τ^{+}(z)}, −τ^{−}(x) − 1 ≤ i ≤ τ^{+}(x). Otherwise the result is clear.

For −τ^{−}(z) − 1 ≤ j ≤ τ^{+}(z), let S_{j} ⊂ B(p, r_{0}) be the smaller sector bounded by rays from the origin to f^{j}z and f^{j+1}z. Let i_{j} be the smallest integer k such that f^{k}x ∈ S_{j}. Thus, |f^{i_{j}}x| = t_{j}|f^{j}z| for some 0 < t_{j} ≤ \frac{1}{2}. By Proposition 2.8, there are at most 2t_{j}^{−2} succesive f^{i}x’s in S_{j}. By Proposition 2.6, each of them satisfies |f^{i}x| ≤ \sqrt{2} t_{j}|f^{j}z|. Thus,

∏_{f^{i}x∈S_{j}}(1 + |f^{i}x|^{2}) ≤ (1 + 2t_{j}^{2}|f^{j}z|^{2})^{[_{\frac{2}{t^{2}_{j}}}]} ≤ e^{4|f^{j}z|^{2}} ≤ 1 + 4β.

It means that m = m(x) ≤ τ^{+}(z) + τ^{−}(z) + 1. Lemma 5.6. There exists D > 0 such that ∀x ∈ S_{r_{0}}^{c},

∏_{i=−τ^{−}(x)}^{τ^{+}(x)}(1 + |f^{−i}x|^{2}) < D,

where τ^{±}(x) = τ_{S_{r_{0}}^{c}}^{±} (x).

Proof. This is a direct corollary of Lemma 5.5.

Now we construct the coordinate systems.

Choose 0 < r̂ < r_{0} such that for any x ∈ S_{r_{0}}^{s}\Ss ; r̂ with f^{τ^{+}(x)+1}(x) ∈ Ss ; r̂, then

∏_{i=−τ^{−}(x)+1}^{τ^{+}(x)−1}(1 − \frac{1}{2}κ′|f^{i}x|^{2}) ≤ \frac{1}{D^{2η′}} ,

where τ^{±}(x) = τ_{S_{r_{0}}^{s} \Ss ; r̂}^{±}(x) and κ′ and η′ are as in Lemma 5.2.ii).

Denote

Γ = {y ∈ B(p, r_{0}) : ∃ − τ_{B(p,r_{0})}^{−}(y) ≤ i ≤ τ_{B(p,r_{0})}^{+}(y) s.t. f^{i}y ∈ Ss ; r̂}.

Choose 0 < r^{∗} < r̂ such that B(p, r^{∗})\W_{r_{0}}^{u}(p) ⊂ Γ. In other words, for any x ∈ B(p, r^{∗})\W_{r_{0}}^{u}(p), if we iterate backwards, then the orbit of x goes to Ss ; r̂ before it leaves B(p, r_{0}). We also assume that r^{∗} is small enough such that ∀x ∉ B(x, r_{0}),

Ĉ_{x}^{u} := ⋃_{y∈B(x,r^{∗}^{3})} π_{x}^{−1}π_{y}C_{y}^{u} and Ĉ_{x}^{s} := ⋃_{y∈B(x,r^{∗}^{3})} π_{x}^{−1}π_{y}C_{y}^{s} are still bounded away from

eath other.

For x ∈ S_{r_{0}}^{s} ∪ S_{r^{∗}/2}^{c} ∪ S_{r_{0}}^{u}, let Ẽ_{x}^{u} ⊕ Ẽ_{x}^{s} coincide with Ξ_{x}^{u} ⊕ Ξ_{x}^{s}. Then we extend these coordinate systems to M\{p} in such a way that Ẽ_{x}^{u} ⊕ Ẽ_{x}^{s} remain C^{1} and with respect to these coordinate systems, ζ_{x} ≤ 1 + η′|x|^{2} for x ∈ Sc ; r̂ and ζ_{x} < 1 for x ∉ B(p, r^{∗}), where

ζ_{x} = min{ζ : |F_{x}σ_{1} − F_{x}σ_{2}| ≤ ζ|σ_{1} − σ_{2}| ∀σ_{1}, σ_{2} ∈ Ĉ_{x}^{u}}.  (5.15)

Since Ẽ_{x}^{u}⊕Ẽ_{x}^{s} are C^{1}, there exists a constant C_{0} > 0 such that ∀x ∈ M\B(p, r^{∗}), y ∈ B(x, r^{∗}^{3}),

|F_{x}σ − F_{y}σ| ≤ \frac{C_{0}}{r^{∗}} d(x, y)  ∀σ ∈ C̃_{x}^{u}.  (5.16)

We may regard this C_{0} as the same as that in (5.2).

6. Local Hölder Condition

In this section we prove that for any x ∈ M\{p}, there exists a neighborhood in which Hölder condition holds with a constant depending on x. We call this property local Hölder condition.

Proposition 6.1. There exist constants H > 0, θ > 0 and r^{∗} > 0, such that for all x ∈ M\{p},

d(E_{x}^{u}, E_{y}^{u}) ≤ \frac{H}{ρ_{x}^{3θ}} d(x, y)^{θ} ∀y ∈ B(x, ρ_{x}^{3}),

where ρ_{x} = min{|x|, r^{∗}}.

We will prove it by showing the following result.

Lemma 6.2. Let {Ẽ_{x}^{u} ⊕ Ẽ_{x}^{s} : x ∈ M} be constructed as in the last section. There exist constants H > 0, θ > 0 and r^{∗} > 0 such that if x ∈ M\{p}, y ∈ B(x, ρ_{x}^{3}), where ρ_{x} = min{|x|, r^{∗}}, then

|σ^{u}(x) − σ^{u}(y)| ≤ \frac{H}{ρ_{x}^{3θ}} d(x, y)^{θ}.

Before we prove the lemma, we give the following fact.

Lemma 6.3. Lemma 6.2 implies Proposition 6.1.

Proof. Recall that π_{y} is a map from the unit circle of T_{y}M to IR ∪ {∞}. By choosing a suitable branch we may think π_{y}^{−1} is well defined. Since C̃^{u} and C̃^{s} are bounded away from each other, restricted to C̃^{u}, both π_{y} and π_{y}^{−1} are Lipschitz with a Lipschitz constant L ∀y ∈ M. Denote e^{u}(y) = π_{y}^{−1}σ^{u}(y).

First, consider the case that x ∈ B(p, r^{∗}). We have

\begin{aligned}d(e^{u}(x), e^{u}(y)) & ≤ L|π_{x}e^{u}(x) − π_{x}e^{u}(y)| \\ & ≤ L(|π_{x}e^{u}(x) − π_{y}e^{u}(y)| + |π_{y}e^{u}(y) − π_{x}e^{u}(y)|). \end{aligned}

Denote σ′ = π_{x}e^{u}(y). We know π_{y}e^{u}(y) − π_{x}e^{u}(y) = π_{y}π_{x}^{−1}σ′ − σ′. By Lemma 3.6 and the definition of π_{x},

\begin{matrix}|π_{y}π_{x}^{−1}σ′ − σ′| = |\frac{h_{21}(x, y) + h_{22}(x, y)σ′}{1 + h_{11}(x, y) + h_{12}(x, y)σ′} − σ′| \\ = & |\frac{h_{21}(x, y) + h_{22}(x, y)σ′ − h_{11}(x, y)σ′ − h_{12}(x, y)σ′^{2}}{1 + h_{11}(x, y) + h_{12}(x, y)σ′} | ≤ C′|Θ(x, y)| \end{matrix}

for some C′ > 0. Since d(x, y) ≤ ρ_{x}^{3} = |x|^{3}, we have

|Θ(x, y)| ≤ \frac{d(x, y)}{|x|} = \frac{d(x, y)^{1−θ}}{|x|^{1−3θ}} · \frac{d(x, y)^{θ}}{|x|^{3θ}} ≤ |x|^{2} · \frac{d(x, y)^{θ}}{ρ_{x}^{3θ}} .

By Lemma 6.2,

|π_{x}e^{u}(x) − π_{y}e^{u}(y)| = |σ^{u}(x) − σ^{u}(y)| ≤ \frac{H}{ρ_{x}^{3θ}} d(x, y)^{θ}.

Thus,

d(e^{u}(x), e^{u}(y)) ≤ L(H + C′|x|^{2})\frac{d(x, y)^{θ}}{ρ_{x}^{3θ}} .

We get the inequality in Proposition 6.1 for x ∈ B(p, r^{∗}) with L(H +C′r^{∗}^{2}) instead of H.

Similarly, since the coordinate systems is C^{1} on M\B(p, r^{∗}), we can find a constant C′′ > 0 such that

|π_{y}π_{x}^{−1}σ′ − σ′| ≤ C′′d(x, y) ≤ C′′r^{∗}^{3θ} \frac{d(x, y)^{θ}}{ρ_{x}^{3θ}} .

The rest of arguments are the same as above.

Proof of Lemma 6.2.

Take θ > 0 such that for x ∈ B(p, r_{0}),

(1 − \frac{3}{4}κ′|x|^{2})(1 + η|x|^{2})^{3θ}(1 + 4η|x|^{2})^{θ} ≤ 1 − \frac{1}{2}κ′|x|^{2} ∀x ∈ B(p, r_{0}), (6.1a)
(1 + \frac{3}{2}η′|x|^{2})(1 + η|x|^{2})^{3θ}(1 + 4η|x|^{2})^{θ} ≤ 1 + 2η′|x|^{2} ∀x ∈ B(p, r_{0}), (6.1b)
\frac{1 + ζ_{x}}{2} ∥Df^{−1}|_{fB(x,ρ_{x}^{3})}∥^{θ} ≤ 1  ∀x ∉ B(p, r^{∗}), (6.1c)

where κ′ and η′ are as in Lemma 5.2.ii), and ζ_{x} is as in (5.15).

Take

H ≥ max\{\frac{4C_{0}}{κ′} , \frac{2C_{0}}{η′} , \frac{2C_{0}r^{∗}^{2}}{1 − ζ_{x}} , 2^{1+θ}B∥Df^{−1}∥^{θ}\},

where C_{0} is as in (5.2) and (5.16) and B is the bound of all σ ∈ C̃^{u} with respect to coordinate systems Ẽ_{x}^{u} ⊕ Ẽ_{x}^{s}.

Define a function s : M → [1, D^{2η′}] by putting

s_{x} = \{ \begin{matrix}max\{1, D^{2η′} ∏_{i=1}^{τ_{B}^{−}(x)}(1 − \frac{κ′}{2} |f^{−i}x|^{2}) \} & x ∈ S_{r_{0}}^{s}\Ss ; r̂; \\ 1, & x ∈ Ss ; r̂; \\ ∏_{i=1}^{τ_{S}^{−}(x)}(1 + 2η′|f^{−i}x|^{2}), & x ∈ Sc ; r̂ ∩ Γ; \\ D^{2η′}, & otherwise, \end{matrix}

where τ_{B}^{−}(x) = τ_{B(p,r_{0})}^{−}(x) and τ_{S}^{−}(x) = τ_{Sc ; r̂∩Γ}^{−}(x).

Let

H = \{σ ∈ C^{u} : |σ^{u}(x) − σ^{u}(y)| ≤ \frac{s_{x}H}{ρ_{x}^{3θ}} d(x, y)^{θ} ∀x ∈ M, y ∈ B(x, ρ_{x}^{3}) \}.

H is closed in the sense that for a sequence {σ_{n}}_{n=1}^{∞} ⊂ H if lim_{n→∞} σ_{n} = σ pointwise, then σ ∈ H. In fact, ∀x ∈ M ∀β > 0, we can find N > 0 such that for all n > N

|σ_{n}(x) − σ(x)| ≤ β.

Thus ∀y ∈ B(x, ρ_{x}^{3}),

|σ_{n}(y) − σ(x)| ≤ |σ_{n}(y) − σ_{n}(x)| + |σ_{n}(x) − σ(x)| ≤ \frac{s_{x}H}{ρ_{x}^{3θ}} d(x, y)^{θ} + β.

Since lim_{n→∞} σ_{n}(y) = σ(y) and β are arbitrary, |σ(y) − σ(x)| ≤ \frac{s_{x}H}{ρ_{x}^{3}} d(x, y)^{θ}. Hence

σ ∈ H.

Let σ ∈ H. Take x ∈ M and y ∈ B(x, ρ_{x}^{3}). We will show that

|(Fσ)(fx) − (Fσ)(fy)| ≤ \frac{s_{fx}H}{ρ_{fx}^{3θ}} d(fx, fy)^{θ}.  (6.2)

If it is true, then we have FH ⊂ H. So by Lemma 5.1 and the closeness of H, for any σ ∈ H, σ^{u} = lim_{n→∞} F^{n}σ ∈ H, which will complete the proof.

We point out here that it is enough to consider the case y ∈ B(x, ρ_{x}^{3}). Because if d(x, y) > ρ_{x}^{3}, then

d(fx, fy) ≥ \frac{d(x, y)}{∥Df^{−1}∥} > \frac{ρ_{x}^{3}}{∥Df^{−1}∥} ≥ \frac{ρ_{fx}^{3}}{2∥Df^{−1}∥}.

So by the choice of H we have \frac{H}{ρ_{fx}^{3θ}} d(fx, fy)^{θ} > \frac{H}{(2∥Df^{−1}∥)^{θ}} ≥ 2B, which means

that (6.2) is always true.

Now suppose x ∈ M\{p} and y ∈ B(x, ρ_{x}^{3}). We have

|(Fσ)(fx) − (Fσ)(fy)| = |F_{x}σ(x) − F_{y}σ(y)|
≤|F_{x}σ(y) − F_{y}σ(y)| + |F_{x}σ(x) − F_{x}σ(y)|.  (6.3)

We consider the following cases.

Case I: x ∈ S_{r_{0}}^{s} ∪ S_{r_{0}}^{u}.

By (6.3) and Lemma 5.2,

|(Fσ)(fx) − (Fσ)(fy)| ≤ \frac{C_{0}d(x, y)}{|x|} + (1 − κ′|x|^{2})|σ(x) − σ(y)|

≤\frac{C_{0}d(x, y)}{|x|} + (1 − κ′|x|^{2})\frac{s_{x}H}{ρ_{x}^{3θ}} d(x, y)^{θ}
=(\frac{C_{0}d(x, y)^{1−θ}ρ_{x}^{3θ}}{|x|} + (1 − κ′|x|^{2})s_{x}H)(\frac{ρ_{fx}}{ρ_{x}} )^{3θ}(\frac{d(x, y)}{d(fx, fy)})^{θ} \frac{1}{ρ_{fx}^{3θ}} d(fx, fy)^{θ}.

Since ρ_{x} ≤ |x|, d(x, y) ≤ |x|^{3} and s_{x}κ′H ≥ 4C_{0}, we have

\frac{C_{0}d(x, y)^{1−θ}ρ_{x}^{3θ}}{|x|} + (1 − κ′|x|^{2})s_{x}H ≤ C_{0}|x|^{2} + (1 − κ′|x|^{2})s_{x}H ≤ (1 − \frac{3}{4}κ′|x|^{2})s_{x}H.

By the definition of s_{x} and ρ_{x}, Corollary 2.3 and 2.4,

(\frac{ρ_{fx}}{ρ_{x}} )^{3θ} ≤ (1 + η|x|^{2})^{3θ}, (\frac{d(x, y)}{d(fx, fy)})^{θ} ≤ (1 + 4η|x|^{2})^{θ}, (1 − \frac{κ′}{2} |x|^{2})s_{x} ≤ s_{fx}.

Therefore, (6.2) follows from (6.1a).

Case II: x ∈ Sc ; r̂ ∩ Γ.

Similarly,

|(Fσ)(fx) − (Fσ)(fy)| ≤ \frac{C_{0}d(x, y)}{|x|} + (1 + η′|x|^{2})|σ(x) − σ(y)|

≤\frac{C_{0}d(x, y)}{|x|} + (1 + η′|x|^{2})\frac{s_{x}H}{ρ_{x}^{3θ}} d(x, y)^{θ}
=(\frac{C_{0}d(x, y)^{1−θ}ρ_{x}^{3θ}}{|x|} + (1 + η′|x|^{2})s_{x}H)(\frac{ρ_{fx}}{ρ_{x}} )^{3θ}(\frac{d(x, y)}{d(fx, fy)})^{θ} \frac{1}{ρ_{fx}^{3θ}} d(fx, fy)^{θ}.

We get

\frac{C_{0}d(x, y)^{1−θ}ρ_{x}^{3θ}}{|x|} + (1 + η′|x|^{2})s_{x}H ≤ C_{0}|x|^{2} + (1 + η′|x|^{2})s_{x}H ≤ (1 + \frac{3}{2}η′|x|^{2})s_{x}H.

Also,

(\frac{ρ_{fx}}{ρ_{x}} )^{3θ} ≤ (1 + η|x|^{2})^{3θ}, (\frac{d(x, y)}{d(fx, fy)})^{θ} ≤ (1 + 4η|x|^{2})^{θ}, (1 + 2η′|x|^{2})s_{x} ≤ s_{fx}.

Therefore, (6.2) follows from (6.1b).

Case III: x ∈ M\(S_{r_{0}}^{s} ∪ Γ ∪ S_{r_{0}}^{u}).

This is the complement of above two cases. In this case ρ_{x} = ρ_{fx} = r^{∗} and s_{x} = s_{fx} = D^{2η′}. By (6.3), (5.16) and (5.15),

|(Fσ)(fx) − (Fσ)(fy)| ≤ \frac{C_{0}d(x, y)}{r^{∗}} + ζ_{x}|σ(x) − σ(y)|

≤\frac{C_{0}d(x, y)}{r^{∗}} + ζ_{x} \frac{s_{x}H}{r^{∗}^{3θ}} d(x, y)^{θ}
=(C_{0}d(x, y)^{1−θ}r^{∗}^{3θ−1} + ζ_{x}s_{x}H)(\frac{d(x, y)}{d(fx, fy)})^{θ} · \frac{1}{r^{∗}^{3θ}} d(fx, fy)^{θ}.

By the choice of H,

C_{0}d(x, y)^{1−θ}r^{∗}^{3θ−1} + ζ_{x}s_{x}H ≤ C_{0}r^{∗}^{2} + ζ_{x}s_{x}H ≤ \frac{1 + ζ_{x}}{2} s_{x}H = \frac{1 + ζ_{x}}{2} s_{fx}H.

Since \frac{d(x, y)}{d(fx, fy)} ≤ ∥Df^{−1}|_{fB(x,ρ_{x}^{3})}∥, by (6.1c) we get (6.2).

7. Distortion Estimates and Proof of Theorem A

The main work in this section is to prove bounded distortion estimates for small unstable curves away from p (Proposition 7.5). This result makes it possible to prove Theorem A through standard arguments. The absolute continuity of stable foliation is also proved (implied in Proposition 7.7) and is then applied to the proof of the corollary. In the end of the section we give a criterion distinguishing between the two cases in Theorem A (Proposition 7.8), which will be used in Part 2.

We fix a rectangle of the form P = [W_{r}^{u}(p), W_{r}^{s}(p)], where r ∈ (0, r^{∗}], such that fP ∪ P ∪ f^{−1}P ⊂ B(p, r^{∗}).

Lemma 7.1. There exist constants H^{+} > 0 and θ > 0 such that if γ ⊂ fP \P is a W^{u}-segment with f^{−i}γ ⊂ P for i = 1, ⋯ n − 1, then for any x, y ∈ γ,

d(E^{u}(f^{−n}x), E^{u}(f^{−n}y)) ≤ H^{+} \frac{d(f^{−n}x, f^{−n}y)^{θ}}{r^{3θ}} .

Proof. We assume θ < \frac{1}{2}. Take H > 0 satisfying Lemma 6.1 and

H ≥ \frac{r^{3θ}C_{β}}{1 − C_{κ}} ,

where C_{β} and C_{κ} are as in Lemma 5.4.

Suppose N is the minimal positive integer such that f^{−N}γ ⊄ P. By the same argument in the proof of Lemma 6.3, it is enough to show that for any x, y ∈ f^{−N}γ, n = 1, ⋯ N,

|σ^{u}(f^{n}x) − σ^{u}(f^{n}y)| ≤ H^{+} \frac{d(f^{n}x, f^{n}y)^{θ}}{r^{3θ}} .

Let 1 ≤ n ≤ N. Fix β ≥ 0 such that ∥Df_{z}∥ ≤ 1 + β ∀ z ∈ B(p, r^{∗}). Take

0 = n_{0} < n_{1} < ⋯ < n_{l} < n_{l+1} = n such that 1 + 2β ≤ ∏_{i=n_{j}}^{n_{j+1}−1}(1 + |f^{i}x|^{2}) ≤ 1 + 4β

∀j = 0, ⋯, l. Then put k_{j} = n_{j+1} − n_{j}, j = 0, ⋯, l.

For j = 0, ⋯, l, define s_{j} = max{1, s̄_{j}}, and

s̄_{j} = \{ \begin{matrix}1, & f^{n_{j}}x ∈ S_{r^{∗}}^{s}; \\ s_{j−1}C_{η} + \frac{r^{3θ}C_{β}}{H} , & f^{n_{j}}x ∈ S_{r^{∗}}^{c}; \\ s_{j−1}, & f^{n_{j}}x ∈ S_{r^{∗}}^{u}, \end{matrix}

where C_{η} and C_{β} are is as in Lemma 5.4. By Lemma 5.5, there are at most m different n_{j} with f^{n_{j}}x ∈ S_{r^{∗}}^{c}. So there are at most m + 1 different s_{j}, and therefore s_{j} are bounded.

The result of the lemma follows with H^{+} = max_{j}{s_{j}H} if we prove that for all

0 ≤ j ≤ l + 1,

|σ^{u}(f^{n_{j}}x) − σ^{u}(f^{n_{j}}y)| ≤ s_{j}H \frac{d(f^{n_{j}}x, f^{n_{j}}y)^{θ}}{r^{3θ}} .  (7.1)

By Lemma 6.2, (7.1) is true for j = 0. Suppose it is true for some 0 ≤ j ≤ l. Recall F_{f^{n_{j}} x}^{k_{j}}σ^{u}(f^{n_{j}}x) = σ^{u}(f^{n_{j+1}}x). We have

|σ^{u}(f^{n_{j+1}}x) − σ^{u}(f^{n_{j+1}}y)| = |F_{f^{n_{j}} x}^{k_{j}}σ^{u}(f^{n_{j}}x) − F_{f^{n_{j}} y}^{k_{j}}σ^{u}(f^{n_{j}}y)|
≤|F_{f^{n_{j}} x}^{k_{j}}σ^{u}(f^{n_{j}}y) − F_{f^{n_{j}} y}^{k_{j}}σ^{u}(f^{n_{j}}y)| + |F_{f^{n_{j}} x}^{k_{j}}σ^{u}(f^{n_{j}}x) − F_{f^{n_{j}} x}^{k_{j}}σ^{u}(f^{n_{j}}y)|.

By Lemma 4.6, d(f^{i}x, f^{i}y) ≤ ηC|f^{i}x|^{3}. So

|f^{n_{j}}x|^{−1} · d(f^{n_{j}}x, f^{n_{j}}y)^{1−θ} ≤ η^{1−θ}C^{1−θ}|f^{n_{j}}x|^{2−3θ)} ≤ 1,

for all f^{n_{j}}x ∈ B(p, r^{∗}) if r^{∗} is small enogh.

Hence by Lemma 5.4.i), we get

|F_{f^{n_{j}} x}^{k_{j}}σ^{u}(f^{n_{j}}y)−F_{f^{n_{j}} y}^{k_{j}}σ^{u}(f^{n_{j}}y)| ≤ C_{β}|f^{n_{j}}x|^{−1}·d(f^{n_{j}}x, f^{n_{j}}y) ≤ C_{β}d(f^{n_{j}}x, f^{n_{j}}y)^{θ}.

To prove (7.1) for j + 1, we need use Lemma 5.4.ii). Consider two cases.

Case I: f^{j}x ∈ S_{r^{∗}}^{s} ∪ S_{r^{∗}}^{u}. Note that in this case, s_{j} = s_{j+1}. By the choice of H,

|σ^{u}(f^{n_{j+1}}x) − σ^{u}(f^{n_{j+1}}y)| ≤ C_{β}d(f^{n_{j}}x, f^{n_{j}}y)^{θ} + s_{j}C_{κ} \frac{H}{r^{3θ}} d(f^{n_{j}}x, f^{n_{j}}y)^{θ}

≤(C_{β} + s_{j}C_{κ} \frac{H}{r^{3θ}} )d(f^{n_{j}}x, f^{n_{j}}y)^{θ} ≤ s_{j+1}H \frac{d(f^{n_{j+1}}x, f^{n_{j+1}}y)^{θ}}{r^{3θ}} .  (7.2)

Case II: f^{j}x ∈ S_{r^{∗}}^{c}. By the definition of s_{j}, we have

|σ^{u}(f^{n_{j+1}}x) − σ^{u}(f^{n_{j+1}}y)| ≤ C_{β}d(f^{n_{j}}x, f^{n_{j}}y)^{θ} + s_{j}C_{η} \frac{H}{r^{3θ}} d(f^{n_{j}}x, f^{n_{j}}y)^{θ}

≤(C_{β} + s_{j}C_{η} \frac{H}{r^{3θ}} )d(f^{n_{j}}x, f^{n_{j}}y)^{θ} ≤ s_{j+1}H \frac{d(f^{n_{j+1}}x, f^{n_{j+1}}y)^{θ}}{r^{3θ}} .  (7.3)

Hence, we get (7.1) for j + 1 instead of j. This completes the proof.

Lemma 7.2. There exist constants H^{−} > 0 and θ > 0 such that if γ ⊂ f^{−1}P \P is a W^{s}-segment with f^{i}γ ⊂ P for i = 1, ⋯ n − 1, then for any x, y ∈ γ,

d(E^{u}(f^{n}x), E^{u}(f^{n}y)) ≤ \frac{H^{−}}{r^{3θ}} d(f^{n}x, f^{n}y)^{θ}.

Proof. The idea is the same as that in last lemma. However, since d(f^{n_{j}}x, f^{n_{j}}y) is decreasing, the fact d(f^{n_{j}}x, f^{n_{j}}y)^{θ} ≤ d(f^{n_{j+1}}x, f^{n_{j+1}}y)^{θ} cannot be used. We need some adjustment to obtain the last inequalities in (7.2) and (7.3).

By Corollary 2.3, we have

\frac{d(f^{n_{j}}x, f^{n_{j}}y)}{d(f^{n_{j+1}}x, f^{n_{j+1}}y)} ≤ ∏_{i=n_{j}}^{n_{j+1}−1}(1 + η|f^{i}x|^{2}) ≤ ∏_{i=n_{j}}^{n_{j+1}−1}(1 + |f^{i}x|^{2})^{η} ≤ (1 + 4β)^{η}.

So if we take θ small enough such that (1 + 4β)^{ηθ}C_{κ} ≤ 1, and then take H ≥

\frac{r^{3θ}(1 + 4β)^{ηθ}C_{β}}{1 − (1 + 4β)^{ηθ}C_{κ}}, and set s_{j} = (s_{j−1}C_{η} + \frac{r^{3θ}C_{β}}{H} )·(1+4β)^{ηθ} for f^{n_{j}}x ∈ S_{r^{∗}}^{c}, then

the last step of (7.2) and (7.3) can go through.

Lemma 7.3. There exists constant \overline{D} > 0 such that if γ ⊂ fP \P is a W^{u}-segment with f^{−i}γ ⊂ P for i = 1, ⋯ n − 1, then ∀x, y ∈ γ,

∑_{i=0}^{n} d(f^{−i}x, f^{−i}y) ≤ \overline{D}.

Proof. By Lemma 4.6,

∑_{i=0}^{n} d(f^{−i}x, f^{−i}y) ≤ C ∑_{i=0}^{n} d(f^{−i}x, f^{−i+1}x).

Choose 0 ≤ n^{u} ≤ n^{s} ≤ n such that f^{−i}x ∈ S_{r^{∗}}^{u} if 0 ≤ i < n^{u}, f^{−i}x ∈ S_{r^{∗}}^{s} if n^{s} < i ≤ n, and f^{i}x ∈ S_{r^{∗}}^{c} if n^{u} ≤ i ≤ n^{s}.

Note that if z ∈ E_{p}^{u} with |z| small, then z ∈ S_{r^{∗}}^{u} and therefore d(z, fz) is close to |fz| − |z|. So we may assume that the sectors S^{u} was chosen in such a way that for all z ∈ S_{r^{∗}}^{u}, d(z, fz) = |fz − z| ≤ K(|fz| − |z|) for some K > 0. Similarly, if z ∈ S_{r^{∗}}^{s}, then d(z, fz) = |fz − z| ≤ K(|z| − |fz|). Thus,

∑_{i=0}^{n^{u}−1} d(f^{−i}x, f^{−i+1}x) ≤ K ∑_{i=0}^{n^{u}−1}(|f^{−i+1}x| − |f^{−i}x|) ≤ K|fx| ≤ Kr^{∗}.

∑_{i=n^{s}+1}^{n} d(f^{−i}x, f^{−i+1}x) ≤ K ∑_{i=n^{s}+1}^{n}(|f^{−i}x| − |f^{−i+1}x|) ≤ K|f^{−n}x| ≤ Kr^{∗}.

By Corollary 2.3 and Lemma 5.6,

∑_{i=n^{u}}^{n^{s}} d(f^{−i}x, f^{−i+1}x) ≤ η ∑_{i=n^{u}}^{n^{s}} |f^{−i}x|^{3} ≤ 2ηr^{∗} log( ∏_{i=n^{u}}^{n^{s}}(1 + |f^{−i}x|^{2})) ≤ 2ηr^{∗} log D.

So the result follows by putting \overline{D} ≥ Cr^{∗}(2η log D + 2K).

For y ∈ W^{u}(x), let d^{u}(x, y) denote the distance between x and y measured along W^{u}(x), and for z ∈ W^{s}(x), let d^{s}(x, z) be defined in an analogous way.

Lemma 7.4. There exists constant I > 0 such that if γ ⊂ fP \P is a W^{u}-segment with f^{−i}γ ⊂ P for i = 1, ⋯ n − 1, then ∀x, y ∈ γ,

log \frac{|Df_{y}^{−n}|_{E_{y}^{u}} |}{|Df_{x}^{−n}|_{E_{x}^{u}} |} ≤ Id^{u}(x, y)^{θ}.

Proof. Denote x_{i} = f^{−i}x, y_{i} = f^{−i}y and γ_{i} = f^{−i}γ.

Choose β ≥ 0 such that ∥Df_{z}∥ ≤ 1 + β ∀ z ∈ B(p, r^{∗}). Take 0 = n_{0} < n_{1} <

⋯ < n_{l} = n such that 1 + 2β ≤ ∏_{i=n_{j−1}+1}^{n_{j}}(1 + |x_{i}|^{2}) ≤ 1 + 4β ∀j = 1, ⋯, l. Then

put k_{j} = n_{j} − n_{j−1}, j = 1, ⋯, l.

Let e(x) = e^{u}(x) be a unit vector in E_{x}^{u}. We have

\frac{|Df_{y}^{−n}|_{E_{y}^{u}} |}{|Df_{x}^{−n}|_{E_{x}^{u}} |} = \frac{|Df_{x_{n}}^{n}e(x_{n})|}{|Df_{x_{n}}^{n}e(y_{n})|} · \frac{|Df_{x_{n}}^{n}e(y_{n})|}{|Df_{y_{n}}^{n}e(y_{n})|}

= ∏_{j=1}^{l}(1 + \frac{|Df_{x_{n_{j}}}^{k_{j}} e(x_{n_{j}}) − Df_{x_{n_{j}}}^{k_{j}} e(y_{n_{j}})|}{|Df_{y_{n_{j}}}^{k_{j}} e(y_{n_{j}})|}) · ∏_{i=1}^{n}(1 + \frac{|Df_{x_{i}}e(y_{i}) − Df_{y_{i}}e(y_{i})|}{|Df_{y_{i}}e(y_{i})|})

≤ ∏_{j=1}^{l}(1 + ∥Df_{x_{n_{j}}}^{k_{j}} ∥ · |e(x_{n_{j}}) − e(y_{n_{j}})|) · ∏_{i=1}^{n}(1 + ∥Df∥d(x_{i}, y_{i})).

By Corollary 2.3, ∥Df_{x_{j}}∥ ≤ 1+3η|x_{j}|^{2}. The choice of n_{i} implies that ∀1 ≤ j ≤ l,

∥Df_{x_{n_{j}}}^{k_{j}} ∥ ≤ ∏_{i=n_{j−1}+1}^{n_{j}}(1 + 3η|x_{i}|^{2}) ≤ ∏_{i=n_{j−1}+1}^{n_{j}}(1 + |x_{i}|^{2})^{3η} ≤ (1 + 4β)^{3η}.

So by Lemma 7.1,

log \frac{|Df_{y}^{−n}|_{E_{y}^{u}} |}{|Df_{x}^{−n}|_{E_{x}^{u}} |} ≤ (1 + 4β)^{3η} \frac{H^{+}}{r^{3θ}} ∑_{j=1}^{l} d(x_{n_{j}}, y_{n_{j}})^{θ} + ∥Df∥ ∑_{i=1}^{n} d(x_{i}, y_{i}). (7.4)

For any z_{i} ∈ γ_{i}, |z_{i}|^{2} > \frac{|x_{i}|^{2}}{2} . Hence ∀1 ≤ j ≤ l,

|Df_{z_{n_{j}}}^{k_{j}} |_{E_{zn_{j}}^{u}} | ≥ ∏_{i=n_{j−1}+1}^{n_{j}}(1 + κ^{u}|z_{i}|^{2}) ≥ ∏_{i=n_{j−1}+1}^{n_{j}}(1 + |x_{i}|^{2})^{\frac{κ^{u}}{2}} ≥ (1 + 2β)^{\frac{κ^{u}}{2}} .

This means that d^{u}(x_{n_{j}}, y_{n_{j}}) and therefore d(x_{n_{j}}, y_{n_{j}}) decrease expomentially as j increases. Thus the first sum in (7.4) is bounded. By Lemma 7.4, the second sum is also bounded. Now it is easy to conclude that

log \frac{|Df_{y}^{−n}|_{E_{y}^{u}} |}{|Df_{x}^{−n}|_{E_{x}^{u}} |} ≤ I_{0}

for some I_{0} > 0 independent of the choice of γ, x and y.

Take γ̄ ⊃ γ in such a way that γ̄ ⊂ fP \P with maximal length. The arguments above tell us that

d^{u}(x_{i}, y_{i}) ≤ e^{2I_{0}} \frac{d^{u}(x̄_{i}, ȳ_{i})}{d^{u}(x̄, ȳ)}d^{u}(x, y),

where x̄_{i} and ȳ_{i} are the extreme points of γ̄_{i}. Note that d(x, y) and d^{u}(x, y) are equivalent for any y ∈ W_{r_{0}}^{u}(x) and that d^{u}(x̄, ȳ) is bounded away from 0 for any given P. Using (7.4) again we get the result.

Proposition 7.5. There exist constants δ > 0 and J_{u} > 1, depending on P, such that if γ is a W^{u}-segment with l(γ) ≤ δ and γ ∩ P = ∅, then ∀x, y ∈ γ and n > 0,

J_{u}^{−1} ≤ \frac{|Df_{y}^{−n}|_{E_{y}^{u}} |}{|Df_{x}^{−n}|_{E_{x}^{u}} |} ≤ J_{u}.

Proof. Ues Lemma 7.4 and the fact that Df_{x}|_{E_{x}^{u}} is uniformly expanding outside P. See the proof in [HY] Proposition 3.1 for more details.

Lemma 7.6. There exist constants δ > 0 and J_{s} > 1, depending on P, such that if γ is a W^{s}-segment with diam(γ) ≤ δ and γ ∩ P = ∅, then ∀x, z ∈ γ and n > 0,

J_{s}^{−1} ≤ \frac{|Df_{z}^{n}|_{E_{z}^{u}} |}{|Df_{x}^{n}|_{E_{x}^{u}} |} ≤ J_{s}.

Proof. By using Lemma 7.2, we can prove a result parallal to Lemma 7.4. The rest is the same as in the proof of the previous proposition.

Proof of Theorem A.

Take a rectangle P as in [HY] Lemma 5.1. Define the first return map g : M\P → M\P. Using the same proof as in [HY] Lemma 5.2, we know that there exists a g-invariant Borel probability measure μ̄ with the property that μ̄ has absolutely continuous conditional measures on the unstable manifolds of f. By pushing forward, we can extend μ̄ to an f-invariant measure on M. That is, let

μ = ∑_{i=0}^{∞} f_{∗}^{i}(μ̄|_{Q_{i}}),  7.5)

where f_{∗}μ̄ = μ̄ ∘ f^{−1}, Q_{0} = M\P and Q_{i} = {x ∈ M\P : fx, ⋯, f^{i}x ∈ P } ∀i ≥ 1. It is easy to see that μ also has absolutely continuous conditional measures on the unstable manifolds of f. So if the series in 7.5) converges, then μM is finite, and after normalization we get an SBR measure of f. Otherwise, μ is a σ-finite measure on M and therefore is an infinite SBR measure of f.

Let Σ_{1} and Σ_{2} be two W^{u}−leaves, and let ι : Σ_{1} → Σ_{2} be a continuous map defined by sliding along the W^{s}−leaves, i.e. for x ∈ Σ_{1}, ι(x) ∈ Σ_{2} ∩ W^{s}(x).

Proposition 7.7. The W^{s}-foliation is locally Lipschitz away from p. More precisely, for any rectangle P = [W_{r}^{u}(p), W_{r}^{s}(p)], there exist constants L > 0 and ϵ > 0 such that ∀x ∈ M\P with [W_{ϵ}^{u}(x), W_{ϵ}^{s}(x)] ∩ P = ∅, and ∀z ∈ W_{ϵ}^{s}(x), ι : W_{ϵ}^{u}(x) → W^{u}(z) is Lipschitz with Lipschitz constant L.

Proof. Let γ be an arbitrarily short segment in W_{ϵ}^{u}(x). We will prove l(ιγ) ≤ L·l(γ) for some L > 0 independent of γ, where l(γ) denote the length of γ.

Denote by γ^{s} the stable curve connecting x_{1} ∈ γ and ιx ∈ W_{ϵ}^{s}(x_{1}). Take n > 0 such that l(f^{n}γ^{s}) ≤ min{l(f^{n}γ), l(f^{n}(ιγ))}. Without loss generality we assume the lengthes of f^{n}γ ∩ P and f^{n}(ιγ) ∩ P are 0 and l(f^{n}γ), l(f^{n}γ^{s}) ≤ δ, where δ is as in Proposition 7.5 and Lemma 7.6. This is possible because otherwise we can take a shorter γ and larger n.

By the continuity of E_{x}^{u} and E_{x}^{s}, there is an L′ > 0 such that l(f^{n}(ιγ)) ≤ L′ · l(f^{n}(γ)). The distorsion estimates in Proposition 7.5 and Lemma 7.6 imply that

l(ιγ) ≤ L′J_{s}J_{u} · l(γ).

Proof of the Corollary of Theorem A.

The Lipschitzness of the W^{s}−foliation is given by Proposition 7.7. This enable us to follow the proof of Lemma 5.3 and Theorem B in [HY] to get the desired results.

The following facts will be used in the proof of Theorem B in the next part.

Take p^{+}, p^{−} ∈ W_{r^{∗}}^{u}(p) in such a way that p^{+} and p^{−} are in different components of W_{r^{∗}}^{u}(p)\{p}, and take q^{+}, q^{−} ∈ W_{r^{∗}}^{s}(p) similarly. We have [q, f^{−i}p^{±}] ∈ W_{r^{∗}}^{u}(q) and lim_{n→∞}[q, f^{−i}p^{±}] = q for q = q^{+} or q^{−}. Put

Δ^{++} = ∑_{n=1}^{∞} d^{u}(q^{+}, [q^{+}, f^{−n}p^{+}])  and  Δ = Δ^{++}+Δ^{+−}+Δ^{−+}+Δ^{−−}, (7.6)

where Δ^{+−}, Δ^{−+} and Δ^{−−} are understood in a similar way as Δ^{++}. It is clear whether Δ < ∞ is independent of the choice of p^{±}. Since W^{s}-foliation is Lipschitz away from p, whether Δ < ∞ is also independent of the choice of q^{±}.

Proposition 7.8. If Δ < ∞, then f admits an SBR measure. Otherwise, f admits an infinite SBR measure.

Proof. We may assume p^{±} and q^{±} are in the boundary of P. Denote by P^{++} the quarter of P bounded by W_{r^{∗}}^{u}(p), W_{r^{∗}}^{s}(p), W_{r^{∗}}^{u}(q^{+}) and W_{r^{∗}}^{s}(p^{+}). It is enough to show that μP^{++} < ∞ if and only if Δ^{++} < ∞.

Put Q_{i}^{++} = {x ∈ Q : fx, ⋯, f^{i}x ∈ P }. By 7.5) μP^{++} = ∑_{i=1}^{∞} μQ_{i}^{++}. Denote

q_{i} = [q^{+}, f^{−i}p^{+}]. Clearly, Q_{i}^{++} is a rectangle whose unstable direction is bounded by W_{r^{∗}}^{s}(q^{+}) and W_{r^{∗}}^{s}(q_{i}). The distortion estimates imply that the densities of the conditional measures of μ on unstable manifolds are bounded away from 0 and ∞ (see e.g. the proof of [HY] Lemma 5.2). So, μQ_{i}^{++} is proportional to d(q^{+}, q_{i}), i.e. there exists J_{u} ≥ 1 such that J_{u}^{−1}d(q^{+}, q_{i}) ≤ μQ_{i}^{++} ≤ J_{u}d(q^{+}, q_{i}) for all i ≥ 1. Now the result becomes clear.

PART 2: Proof of Theorem B

In this part we assume that Assumption A holds. Therefore f can be expressed as (1.3) and (1.4) in some neighborhood B(p, r) of the fixed point p.

Take z = (x, y) ∈ B(p, r) and denote z̄ = (x̄, ȳ) = [z, fz]. To determine whether the series in (7.6) converges, we should estimate the order of x̄ − x as x → 0. Since both z̄ and fz are in W_{r}^{s}(fz), we need to know the slope of W_{r}^{s}(fz). Let v_{z}^{s} be a real number or ∞ such that (v_{z}^{s}, 1) is a tangent vector of W_{r}^{s}(z). Most of the work in this part is estimating v_{z}^{s} for z near y−axis.

8. Preliminaries

We assume that v_{z}^{s} has form −ρ\frac{x}{y}, where ρ = ρ(x, y). Results in Lemma 8.3 and Proposition 8.4 imply that ρ(x, y) = \frac{a_{2}}{b_{2}} + ρ̂(y) + R_{ρ}(x, y), where ρ̂(y) is determined by equations (8.6) and (8.7) in Proposition 8.4, and R_{ρ}(x, y) → 0 as x → 0. This facts allow us to estimate the bounds of ρ(x, y) in §9 and §10.

By (1.3) we know that for z = (x, y) ∈ B(p, r),

Df_{z} = ( \begin{matrix}1 + ϕ(x, y) + xϕ_{x}(x, y) & xϕ_{y}(x, y) \\ −yψ(x, y) & 1 − ψ(x, y) − yψ_{y}(x, y), \end{matrix} )  (8.1)

where ϕ_{x} denotes the partial derivative of ϕ with respect to x, etc. The nondegeneracy condition (1.1) implies that on B(p, r), ϕ(x, y) + xϕ_{x}(x, y) ≥ κ^{u}|(x, y)|^{2} and ψ(x, y) + yψ_{y}(x, y) ≥ κ^{s}|(x, y)|^{2}. In other words, for all z = (x, y) ∈ B(p, r),

3a_{0}x^{2} +2a_{1}xy +a_{2}y^{2} ≥ κ^{u}(x^{2} +y^{2}),  b_{0}x^{2} +2b_{1}xy +3b_{2}y^{2} ≥ κ^{s}(x^{2} +y^{2}). (8.2)

Therefore

a_{0}, a_{2}, b_{0}, b_{2} > 0.  (8.3)

In this part we always assume that z = (x, y) ∈ B(p, r). we also assume that r is small enough such that all higher order terms can be controlled by corresponding lower order terms. We will denote z_{n} = (x_{n}, y_{n}) = f^{n}z, ϕ_{n} = ϕ(x_{n}, y_{n}), ψ_{n} = ψ(x_{n}, y_{n}), etc. Without loss generality we only consider the case that z is in the first quarter. So we have x_{n}, y_{n} ≥ 0 except for opposite statement. Also we may assume

−1 ≤ v_{z}^{s} ≤ 1 ∀z ∈ B(p, r). (8.4)

Otherwise we can rescale x-axis and y-axis by constant factors. This does not change the conditions of Theorem B because by (1.3) and (1.4) the ratio of a_{i}/b_{i} and the signs of a_{i} and b_{i}, i = 0, 1, 2, remain the same.

Lemma 8.1. Let {t_{n}} be a sequence of positive numbers, and let C > 0 and α > 0. i) If ∀n ≥ 1, t_{n−1} ≥ t_{n} + Ct_{n}^{1+α}, then there exist D > 0, k_{0} ≥ 1 such that

t_{n} ≤ D(n − k_{0})^{− \frac{1}{α}} for all large n. Therefore ∑_{n=1}^{∞} t_{n} converges as α < 1.

ii) If ∀n ≥ 1, t_{n−1} ≤ t_{n} + Ct^{1+α}, then there exist D > 0, k_{0} ≥ 1 such that

t_{n} ≥ D(n + k_{0})^{− \frac{1}{α}} for all large n. Therefore ∑_{n=1}^{∞} t_{n} diverges as α ≥ 1.

Proof. i) Take D > 0 such that αCD^{α} ≥ 2. Suppose t_{n} ≥ Dk^{− \frac{1}{α}} for some integer k > 1. Then

t_{n−1} ≥ Dk^{− \frac{1}{α}} (1 + C · (Dk^{− \frac{1}{α}})^{α}) = Dk^{− \frac{1}{α}} (1 + CD^{α}k^{−1}).

It is easy to see that if k is large enough , then (1 − k^{−1})(1 + CD^{α}k^{−1})^{α} > 1 or k^{− \frac{1}{α}} (1 + CD^{α}k^{−1}) ≥ (k − 1)^{− \frac{1}{α}}. So t_{n−1} ≥ D(k − 1)^{− \frac{1}{α}}.

The arguments show that if t_{n−1} ≤ D(n − k_{0} − 1)^{− \frac{1}{α}} for some k_{0} ≥ 1, then t_{n} ≤ D(n − k_{0})^{− \frac{1}{α}}.

ii) The proof is similar.

Lemma 8.2. Let z = (0, y) and z_{n} = (0, y_{n}) = f^{n}z. Then ∑_{n=0}^{∞} y_{n}^{3} ≤ \frac{2}{b_{2}} y_{0} and the

convergence is uniform for all y_{0} ∈ [0, r].

Proof. Since y_{n+1} = y_{n}−y_{n}ψ(0, y_{n}), (1.3) and (1.4) imply y_{n}−y_{n+1} = y_{n}ψ(0, y_{n}) ≥ \frac{b_{2}}{2} y_{n}^{3}. So

∑_{n=0}^{∞} y_{n}^{3} ≤ \frac{2}{b_{2}} ∑_{n=0}^{∞}(y_{n} − y_{n+1}) = \frac{2}{b_{2}} y.

The uniformity is clear.

Denote z_{1} = fz and ρ_{1} = ρ(z_{1}). Note that ρ_{1} = ρ(x(1 + ϕ), y(1 − ψ)) is also a function of z = (x, y). Let

Δ_{ρ}(x, y) =(ρ − ρ_{1})(1 + ϕ)(1 − ψ) + ρ_{1}y(1 + ϕ)ψ_{y} − y(1 − ψ)ϕ_{y}
−ρ_{1}ρx(1 + ϕ)ψ_{x} + ρx(1 − ψ)ϕ_{x}.  (8.5)

Lemma 8.3. If v_{z}^{s} ≤ −ρ(z)\frac{x}{y} and 0 ≤ Δ_{ρ}(x, y), then v_{z_{1}}^{s} ≤ −ρ(z_{1})\frac{x_{1}}{y_{1}} . The result

also holds if all “≤” are replaced by “≥”.

Proof. Since the map v → \frac{(1 + c_{11})v + c_{12}}{c_{21}v + (1 − c_{12})} is increasing as c_{ij} small, by (8.1),

v_{z_{1}}^{s} ≤ \frac{−(1 + ϕ + xϕ_{x}) · ρx + xϕ_{y} · y}{−yψ_{x} · ρx + (−ψ − yψ_{y}) · y}

To get the result, use the fact that 0 ≤ Δ_{ρ}(x, y) is equivalent to

\frac{−(1 + ϕ + xϕ_{x}) · ρx + xϕ_{y} · y}{−yψ_{x} · ρx + (1 − ψ − yψ_{y}) · y} ≤ −ρ_{1} \frac{x(1 + ϕ)}{y(1 − ψ)}.

Next proposition plays a key role for the proof of Theorem B.

Proposition 8.4. There exist a Lipschitz function ρ̂ on [0, r] with ρ̂(0) = 0 satisfying the following two equations

Δ_{\frac{a_{2}}{b_{2}} +ρ̂}(0, y) =(ρ̂(y) − ρ̂(y_{1}^{(0)}))(1 + ϕ)(1 − ψ)
+(\frac{a_{2}}{b_{2}} + ρ̂(y_{1}^{(0)}))y(1 + ϕ)ψ_{y} − y(1 − ψ)ϕ_{y} = 0  (8.6)

and

b_{2} log(1 + ϕ) + a_{2} log(1 − ψ) − b_{2} ∫_{y_{1}^{(0)}}^{y} \frac{ρ̂(t)}{t}dt = 0,  (8.7)

where ϕ = ϕ(0, y), ψ = ψ(0, y) and y_{1}^{(0)} = y(1 − ψ(0, y)).

Proof. Denote z = (0, y), z_{n} = f^{n}z = (0, y_{n}), ϕ_{n} = ϕ(0, y_{n}) etc. So y_{1}^{(0)} can be written as y_{1}. Define ρ̂(0) = 0 and for y ≠ 0 define

ρ̂(y) = ∑_{n=0}^{∞} \frac{b_{2}y_{n}(1 − ψ_{n})ϕ_{y}(0, y_{n}) − a_{2}y_{n}(1 + ϕ_{n})ψ_{y}(0, y_{n})}{b_{2}(1 + ϕ_{n})(1 − ψ_{n})} ∏_{k=0}^{n−1}(1 − \frac{y_{k}ψ_{y}(0, y_{k})}{1 − ψ_{k}}).

(8.8)

By (1.4) b_{2}y_{n}ϕ_{y}(0, y_{n}) − a_{2}y_{n}ψ_{y}(0, y_{n}) = O(y_{n}^{3}). The product in each term in (8.8) is less than 1 because yψ_{y} > 0. So by Lemma 8.2,

|ρ̂(y)| ≤ Lρ̂y  (8.9)

for some Lρ̂ > 0.

It is easy to check

ρ̂(y_{1}) = ∑_{n=1}^{∞} \frac{b_{2}y_{n}(1 − ψ_{n})ϕ_{y}(0, y_{n}) − a_{2}y_{n}(1 + ϕ_{n})ψ_{y}(0, y_{n})}{b_{2}(1 + ϕ_{n})(1 − ψ_{n})} ∏_{k=1}^{n−1}(1 − \frac{y_{k}ψ_{y}(0, y_{k})}{1 − ψ_{k}}).

Therefore,

ρ̂(y) = (1 − \frac{yψ_{y}(0, y)}{1 − ψ})ρ̂(y_{1}) + \frac{b_{2}y(1 − ψ)ϕ_{y}(0, y) − a_{2}y(1 + ϕ)ψ_{y}(0, y)}{b_{2}(1 + ϕ)(1 − ψ)}.

Multiplying by (1 + ϕ)(1 − ψ), we get (8.6).

Note y_{1} = y(1 − ψ). Differentiating the left hand side of (8.7) with respect to y, we get

\frac{b_{2}ϕ_{y}}{1 + ϕ} − \frac{a_{2}ψ_{y}}{1 − ψ} + b_{2} \frac{ρ̂(y)}{y} − b_{2} \frac{ρ̂(y_{1})}{y(1 − ψ)}(1 − ψ − yψ_{y}).

By (8.6) it is equal to 0. This means that the left hand side of (8.7) is a constant. Since ρ̂(y), ϕ(0, y), ψ(0, y) → 0 as y → 0, this constant must be zero.

Now we prove Lipschitzness of ρ̂. We will prove that the derivative of ρ̂ is bounded by a constant for y ≠ 0. This combining with (8.9) implies Lipschitzness of ρ̂ on [0, r].

By (1.4) we can write

\frac{b_{2}y_{n}(1 − ψ_{n})ϕ_{y}(0, y_{n}) − a_{2}y_{n}(1 + ϕ_{n})ψ_{y}(0, y_{n})}{b_{2}(1 + ϕ_{n})(1 − ψ_{n})} = C′y_{n}^{3} + o(y_{n}^{3})

and

1 − \frac{y_{k}ψ_{y}(0, y_{k})}{1 − ψ_{k}} = (1 − ψ_{k})^{2} \frac{1 − ψ_{k} − y_{k}ψ_{y}(0, y_{k})}{(1 − ψ_{k})^{3}} = (1 − ψ_{k})^{2}(1 + C′′y_{k}^{3} + o(y_{k}^{3}))

for some constants C′ and C′′. Clearly, ∏_{k=0}^{n−1}(1 − ψ_{k}) = \frac{y_{n}}{y}. Hence we can write

(8.8) as

\begin{matrix}ρ̂(y) = & ∑_{n=0}^{∞}[C′y_{n}^{3} + o(y_{n}^{3})] ∏_{k=0}^{n−1}(1 − ψ_{k})^{2}(1 + C′′y_{k}^{3} + o(y_{k}^{3})) \\ = & ∑_{n=0}^{∞} \frac{C′y_{n}^{5} + o(y_{n}^{5})}{y^{2}} ∏_{k=0}^{n−1}(1 + C′′y_{k}^{3} + o(y_{k}^{3})). \end{matrix}

Since \frac{dy_{k+1}}{dy_{k}} = 1 − ψ_{k} − y_{k}ψ_{y}(0, y_{k}) ≤ 1 − ψ_{k} = \frac{y_{k+1}}{y_{k}}, we get \frac{dy_{k}}{dy} ≤ \frac{y_{k}}{y} ≤ 1.

Thus,

|\frac{d}{dy} \frac{C′y_{n}^{5} + o(y_{n}^{5})}{y^{2}} | ≤ C_{1} \frac{y_{n}^{5}}{y^{3}} ≤ C_{1} \frac{y_{n}^{3}}{y}

for some C_{1} > 0. By Lemma 8.2, ∏_{k=0}^{n−1}(1 + C′′y_{k}^{3} + o(y_{k}^{3})) ≤ C_{2}. Hence,

|\frac{d}{dy} ∏_{k=0}^{n−1}(1 + C′′y_{k}^{3} + o(y_{k}^{3}))|

=|∑_{k=0}^{n−1} \frac{3C′′y_{k}^{2} + o(y_{k}^{2})}{1 + C′′y_{k}^{3} + o(y_{k}^{3})} · \frac{y_{k}}{y} · ∏_{k=0}^{n−1}(1 + C′′y_{k}^{3} + o(y_{k}^{3}))|

≤ ∑_{k=0}^{n−1} 4|C′′|y_{k}^{2} · \frac{y_{k}}{y} · C_{2} ≤ \frac{4|C′′|C_{2}}{y} ∑_{k=0}^{n−1} y_{k}^{3} ≤ C_{3}.

From these inequalities we get

|∑_{n=0}^{∞} \frac{d}{dy} [\frac{C′y_{n}^{5} + o(y_{n}^{5})}{y^{2}} ∏_{k=0}^{n−1}(1 + C′′y_{k}^{3} + o(y_{k}^{3}))] | ≤ ∑_{n=0}^{∞}(C_{1} \frac{y_{n}^{3}}{y} · C_{2} + 2C′y_{n}^{3} · C_{3}).

Now it is easy to see by Lemma 8.2 that the convergence is uniform. Thus, we

know that |\frac{dρ̂(y)}{dy}| is bounded by \frac{2}{b_{2}} (C_{1}C_{2} + 2C′C_{3}r) for all y ∈ (0, r]. This proves

the result.

9. Existence of SBR Measures

In this section we prove the first part of Theorem B. We estimate the upper bounds of |v_{z}^{s}| in Lemma 9.1 and then show that the general terms of the serieses in (7.6) decrease at a rate faster than n^{− \frac{1}{α}} (implied in Lemma 9.2).

Lemma 9.1. Suppose αa_{2} > 2b_{2}, a_{1} = 0 = b_{1} and a_{0}b_{2} − a_{2}b_{0} > 0. Then for any point q = (0, y_{q}) with y_{q} > 0 small, there exists ϵ > 0 such that ∀z_{0} = (x_{0}, y_{0}) ∈ W_{ϵ}^{u}(q) with x_{0} > 0,

v_{z_{0}}^{s} ≥ −( \frac{a_{2}}{b_{2}} + ρ̂(y_{0}))(1 − x^{α})\frac{x_{0}}{y_{0}} .  (9.1)

Proof. For each z_{0} = (x_{0}, y_{0}) ∈ W_{r}^{u}(q), z_{n} = (x_{n}, y_{n}) = f^{n}z_{0}, we denote

ρ_{i} = ρ(z_{i}) = (\frac{a_{2}}{b_{2}} + ρ̂(y))(1 − x_{i}^{α})  ∀i ≥ 0.  (9.2)

We show that for y_{q} > 0 small, there exists ϵ > 0 such that ∀z_{0} = (x_{0}, y_{0}) ∈ W_{ϵ}^{u}(q) with x_{0} > 0,

v_{z_{i}}^{s} ≤ −ρ_{i} \frac{x_{i}}{y_{i}} and ρ_{i}x_{i} ≤ y_{i}

imply

v_{z_{i+1}}^{s} ≤ −ρ_{i+1} \frac{x_{i+1}}{y_{i+1}} .

SBR MEASURES FOR “ALMOST ANOSOV” DIFFEOMORPHISMS 35

This proves the result of the lemma. In fact, if (9.1) is not true, then for some large n, ρ_{n}x_{n} ≥ y_{n} and therefore v_{z_{n}}^{s} ≤ −ρ_{n} · \frac{x_{n}}{y_{n}} < −1, contradicting to (8.4).

By Lemma 8.3, it is enough to show for such z_{i} = (x_{i}, y_{i}),

Δ_{ρ_{i}}(x_{i}, y_{i}) = (ρ_{i} − ρ_{i+1})(1 + ϕ_{i})(1 − ψ_{i})
+ ρ_{i+1}y_{i}(1 + ϕ_{i})ψ_{y}(x_{i}, y_{i}) − y_{i}(1 − ψ_{i})ϕ_{y}(x_{i}, y_{i})
− ρ_{i}ρ_{i+1}x_{i}(1 + ϕ_{i})ψ_{x}(x_{i}, y_{i}) + ρ_{i}x_{i}(1 − ψ_{i})ϕ_{x}(x_{i}, y_{i}) ≥ 0. (9.3)

Note that x_{i+1}^{α} − x_{i}^{α} = x_{i}^{α}(1 + ϕ_{i})^{α} − x_{i}^{α} = αx_{i}^{α}(ϕ_{i} + O(ϕ_{i}^{2})) = αa_{2}x_{i}^{α}y_{i}^{2} + x_{i}^{α}O(x_{i}^{2} + y_{i}^{3}). By (9.2) we have

\begin{matrix}ρ_{i} − ρ_{i+1} = & (ρ̂(y_{i}) − ρ̂(y_{i+1}))(1 − x_{i+1}^{α}) + (\frac{a_{2}}{b_{2}} + ρ̂(y_{i}))(x_{i+1}^{α} − x_{i}^{α}) \\ = & ρ̂(y_{i}) − ρ̂(y_{i+1}) + \frac{a_{2}}{b_{2}} αa_{2}x_{i}^{α}y_{i}^{2} + x_{i}^{α}O(x_{i}^{2} + y_{i}^{3}). \end{matrix}

Since a_{1} = 0 = b_{1}, by (1.4) we see that ϕ_{y}(x, y) = ϕ_{y}(0, y)+xO(x+y), ψ_{y}(x, y) = ψ_{y}(0, y) + xO(x + y) = 2b_{2}y + O(x^{2} + y^{2}). So

\begin{matrix}ρ_{i+1}y_{i}ψ_{y}(x_{i}, y_{i}) = & (\frac{a_{2}}{b_{2}} + ρ̂(y_{i+1}))y_{i}(ψ_{y}(0, y_{i})+x_{i}O(x_{i}+y_{i})) \\ − & \frac{a_{2}}{b_{2}} x_{i}^{α}(2b_{2}y_{i}^{2} + y_{i}O(x_{i}^{2}+y_{i}^{2})), \end{matrix}

and

y_{i}ϕ_{y}(x_{i}, y_{i}) = y_{i}(ϕ_{y}(0, y_{i}) + x_{i}O(x_{i} + y_{i})).

By Lemma 8.4,

\begin{matrix}(ρ̂(y_{i}) − ρ̂(y_{i+1}^{(0)}))(1 + ϕ_{i})(1 − ψ_{i}) \\ + & (\frac{a_{2}}{b_{2}} + ρ̂(y_{i+1}^{(0)}))y_{i}(1 + ϕ_{i})ψ_{y}(x_{i}, y_{i}) − y_{i}(1 − ψ_{i})ϕ_{y}(x_{i}, y_{i}) = 0, \end{matrix}

where y_{i+1}^{(0)} = y_{i}(1−ψ(0, y_{i})). Note that y_{i+1} = y_{i}(1−ψ(x_{i}, y_{i})). So both y_{i+1}−y_{i+1}^{(0)} and ρ̂(y_{i+1}) − ρ̂(y_{i+1}^{(0)}) have order O(x_{i}^{2}y_{i}). Hence

(ρ_{i} − ρ_{i+1})(1 + ϕ_{i})(1 − ψ_{i}) + ρ_{i+1}y_{i}(1 + ϕ_{i})ψ_{y}(x_{i}, y_{i}) − y_{i}(1 − ψ_{i})ϕ_{y}(x_{i}, y_{i})

=\frac{a_{2}}{b_{2}} (αa_{2} − 2b_{2})x_{i}^{α}(y_{i}^{2} + O(x_{i}^{2} + y_{i}^{3})) + x_{i}y_{i}O(x_{i} + y_{i}).

Also, since ψ_{x}(x, y) = 2a_{0}x + O(x^{2} + y^{2}) and ψ_{x}(x, y) = 2b_{0}x + O(x^{2} + y^{2}), −ρ_{i}ρ_{i+1}x_{i}ψ_{x}(x_{i}, y_{i}) + ρ_{i}x_{i}ϕ_{x}(x_{i}, y_{i}) = 2ρ_{i}(−ρ_{i+1}b_{0} + a_{0})x_{i}^{2} + x_{i}O(x_{i}^{2} + y_{i}^{2}).

By (9.2), we have

−ρ_{i+1}b_{0} + a_{0} = \frac{a_{0}b_{2} − a_{2}b_{0}}{b_{2}} + \frac{a_{2}}{b_{2}} x_{i+1}^{α} + O(y_{i+1}).

36 HUYI HU

So

−ρ_{i}ρ_{i+1}x_{i}ψ_{x}(x_{i}, y_{i})+ρ_{i}x_{i}ϕ_{x}(x_{i}, y_{i}) = 2ρ_{i}(\frac{a_{0}b_{2} − a_{2}b_{0}}{b_{2}} + \frac{a_{2}}{b_{2}} x_{i+1}^{α})x_{i}^{2}+x_{i}O(x_{i}^{2}+y_{i}^{2}).

Therefore, (9.3) can be written as

\begin{matrix}Δ_{ρ_{i}}(x_{i}, y_{i}) = & \frac{a_{2}}{b_{2}} (αa_{2} − 2b_{2})x_{i}^{α}(y_{i}^{2} + O(x_{i}^{2} + y_{i}^{3})) \\ + & 2ρ_{i}(\frac{a_{0}b_{2} − a_{2}b_{0}}{b_{2}} + \frac{a_{2}}{b_{2}} x_{i+1}^{α})x_{i}^{2} + x_{i}O(x_{i}^{2} + y_{i}^{2}). \end{matrix}

Now the assumptions a_{2} > 4b_{2} and a_{0}b_{2} − a_{2}b_{0} > 0 imply Δ_{ρ}(x_{i}, y_{i}) ≥ 0 if y_{0} and x_{0} are small. Lemma 9.2. Let z_{0} = (x_{0}, y_{0}) with x_{0}, y_{0} > 0. If for all z = (x, y) in the stable curve that joins z̄_{0} and z_{1},

v_{z}^{s} ≥ −( \frac{a_{2}}{b_{2}} + ρ̂(y))(1 − x^{α})\frac{x}{y},  (9.4)

then

x̄_{0} ≥ x_{0} + Cx_{0}^{1+α}  (9.5)

for some constant C = C(y_{q}) > 0. Proof. We regard W_{r}^{s}(x) as the graph of a function x = x(y). Then (9.6) gives

\frac{dx}{dy} ≥ −(\frac{a_{2}}{b_{2}} + ρ̂(y))(1 − x^{α})\frac{x}{y},

i.e.

\frac{dx}{x(1 − x^{α})} + (\frac{a_{2}}{b_{2}} + ρ̂(y))\frac{dy}{y} ≥ 0.

Intergrating it from z_{1} = (x_{1}, y_{1}) to z̄_{0} = (x̄_{0}, ȳ_{0}), we get

log \frac{x̄_{0}}{x_{1}} − \frac{1}{α} log \frac{1 − x̄_{0}^{α}}{1 − x_{1}^{α}} + \frac{a_{2}}{b_{2}} log \frac{ȳ_{0}}{y_{1}} + ∫ ȳ_{0} ; y_{1} \frac{ρ̂(y)}{y}dy ≥ 0.

From now on we omit subscript 0 in the rest of proof. This inequality implies

x̄ ≥ x_{1}(\frac{1 − x̄^{α}}{1 − x_{1}^{α}} )^{\frac{1}{α}} (\frac{y_{1}}{ȳ} )^{\frac{a_{2}}{b_{2}}} exp[− ∫ ȳ ; y_{1} \frac{ρ̂(y)}{y}dy].

Since x_{1} = x(1 + ϕ) and y_{1} = y(1 − ψ), we can write

\frac{x̄}{x} ≥(1 + ϕ)(1 − ψ)^{\frac{a_{2}}{b_{2}}} (\frac{1 − x̄^{α}}{1 − x_{1}^{α}} )^{\frac{1}{α}} (\frac{y}{ȳ})^{\frac{a_{2}}{b_{2}}} exp[− ∫ ȳ ; y_{1} \frac{ρ̂(y)}{y}dy]
= exp \frac{1}{b_{2}}\{b_{2} log(1 + ϕ) + a_{2} log(1 − ψ) − b_{2}∫_{y_{1}}^{y} \frac{ρ̂(y)}{y}dy\}
·(\frac{1 − x̄^{α}}{1 − x_{1}^{α}} )^{\frac{1}{α}} (\frac{y}{ȳ})^{\frac{a_{2}}{b_{2}}} exp ∫ y ; ȳ \frac{ρ̂(y)}{y}dy.

The first factor on the right side is of the form 1 + O(x), because by Lemma 8.4 and (8.9),

b_{2} log(1 + ϕ) + a_{2} log(1 − ψ) − b_{2} ∫_{y_{1}}^{y} \frac{ρ̂(y)}{y}dy = b_{2} ∫_{y_{1}^{(0)}}^{y_{1}} \frac{ρ̂(y)}{y}dy = O(x),

where y_{1}^{(0)} = y(1 − ψ(0, y)).

Note that x_{1} = x + xϕ. We have

\frac{1 − x̄^{α}}{1 − x_{1}^{α}} = 1 + \frac{x^{α}(1 + ϕ)^{α} − x̄^{α}}{1 − x_{1}^{α}} = 1 + \frac{x^{α} + αx^{α}ϕ − x̄^{α} + O(ϕ^{2})}{1 − x_{1}^{α}}.

Without loss generality, we may assume x̄^{α} − x^{α} ≤ \frac{1}{2}x^{α}ϕ, otherwise we have

\frac{x̄}{x} > (1 + \frac{1}{2}ϕ)^{\frac{1}{α}} ≥ 1 + Cx^{α} for some C = O(y^{2}) > 0 and therefore (9.5) is true. Thus, we get

(\frac{1 − x̄^{α}}{1 − x_{1}^{α}} )^{\frac{1}{α}} ≥ (1 + \frac{αx^{α}ϕ + O(ϕ^{2})}{2(1 − x_{1}^{α})})^{\frac{1}{α}} ≥ 1 + \frac{1}{4}x^{α}ϕ.

Since both z̄ = (x̄, ȳ) and z = (x, y) are in the same local unstable manifold, |ȳ − y| ≤ N(x̄ − x) ≤ N(x_{1} − x) = Nxϕ for some N > 0. So,

|(\frac{y}{ȳ})^{\frac{a_{2}}{b_{2}}}| ≤ (1 + \frac{Nxϕ}{ȳ})^{\frac{a_{2}}{b_{2}}} = 1 + O(x).

By (8.9) we have

exp ∫ y ; ȳ \frac{ρ̂(y)}{y}dy ≤ exp(Lρ̂|y − ȳ|) ≤ exp(Lρ̂Nxϕ) = 1 + O(x).

Summarizing these results, we get that if x is small enough, then

\frac{x̄}{x} ≥ 1 + \frac{1}{4}x^{α}ϕ + O(x) ≥ 1 + \frac{a_{2}y^{2}}{8} x^{α}.

Proof of Theorem B (I).

By Lemma 9.1, 9.2 and 8.1.i) we know that the sum Δ^{++} in (7.6) converges. Similary, we have Δ^{−+}, Δ^{−−}, Δ^{+−} ≤ ∞. Thus the result follows from Proposition 7.8.

Remark 9.3. By Lemma 9.2 and 8.1.i), we can see that under the assumptions of the theorem, the rate of convergence of the series in (7.6) is faster than n^{− \frac{1}{α}}.

10. Existence of Infinite SBR Measures

We prove the second part of Theorem B. We get the lower bounds of |v_{z}^{s}| in Lemma 10.2 and then prove in Lemma 10.3 that there is at least one series in (7.6) whose general terms decrease at a rate slower than n^{− \frac{1}{2}} .

Lemma 10.1. Suppose 2a_{2} < αb_{2}. Then for any constants K, N > 0, point q = (0, y_{q}) with y_{q} > 0 small, there exists ϵ > 0 such that ∀z_{0} = (x_{0}, y_{0}) ∈ W_{ϵ}^{u}(q) with x_{0} > 0, the inequalities

x_{0}y_{0} ∏_{j=0}^{n}(1 − \frac{7}{8}y_{i}ψ_{y}(0, y_{i}))^{−1} ≥ N  and  Kx_{n} ≤ y_{n}

hold simultaneously for some n = n(z_{0}) > 0.

Proof. We may assume that K is large enough such that if Kx ≤ y, then

1 − \frac{7}{8}yψ_{y}(0, y) ≤ 1 − \frac{13}{8}b_{2}y^{2} ≤ (1 − ψ)^{\frac{3}{2}}

and

(1 + ϕ)^{2}(1 − ψ)^{α_{+}} ≤ 1

for some α < α_{+} < 1. Hence for any z_{0} = (x_{0}, y_{0}) with Kx_{0} < y_{0}, we have x_{1}^{2}y_{1}^{α_{+}} ≤ x_{1}^{2}(1 + ϕ_{1})^{2}y_{1}^{α_{+}}(1 − ψ_{1})^{α_{+}} ≤ x_{0}^{2}y_{0}^{α_{+}}. Let n = n(z_{0}) be the largest integer such that Kx_{n} ≤ y_{n}. Thus Kx_{n+1} > y_{n+1}. So we have

x_{0}^{2}y_{0}^{α_{+}} ≥ x_{n+1}^{2}y_{n+1}^{α_{+}} ≥ K^{−2}y_{n+1}^{2+α_{+}}

and therefore

\frac{x_{0}y_{0}}{∏_{j=0}^{n}(1 − \frac{7}{8}y_{i}ψ_{y}(0, y_{i}))} ≥ \frac{y0^{\frac{3}{2}} · x_{0}y_{0}}{y0^{\frac{3}{2}} · ∏_{j=0}^{n}(1 − ψ_{j})^{\frac{3}{2}}} ≥ \frac{x_{0}y0^{\frac{5}{2}}}{yn+1^{\frac{3}{2}}} ≥ \frac{y_{0}^{(5−α_{+})/2}}{Ky_{n+1}^{(1−α_{+})/2}} .

Since y_{n+1} → 0 as z_{0} = (x_{0}, y_{0}) → q = (0, y_{q}), we get the result.

Lemma 10.2. Suppose 2a_{2} < αb_{2} and b_{1} < 0 < a_{1}. Then for any point q = (0, y_{q}) with y_{q} > 0, there exists ϵ > 0 such that ∀z_{0} = (x_{0}, y_{0}) ∈ W_{ϵ}^{u}(q) with x_{0} > 0,

v_{z_{0}}^{s} ≤ −( \frac{a_{2}}{b_{2}} + ρ̂(y_{0}))\frac{x_{0}}{y_{0}} .  (10.1)

Proof. For each z_{0} = (x_{0}, y_{0}) ∈ W_{r}^{u}(q), define c_{0} = 0 and

c_{i} = \frac{a_{2}}{b_{2}} (a_{1} − \frac{a_{2}}{b_{2}} b_{1})x_{0}y_{0} ∏_{j=1}^{i−1}(1 − \frac{7}{8}y_{i}ψ_{y}(0, y_{i}))^{−1}  ∀i > 0.  (10.2)

We have

c_{i+1} − c_{i} = c_{i+1} · \frac{7}{8}y_{i}ψ_{y}(0, y_{i})  ∀i > 0.  (10.3)

Put

ρ_{i} = max\{ −\frac{1}{3}, \frac{a_{2}}{b_{2}} + ρ̂(y_{i}) − c_{i}\}  ∀i ≥ 0.  (10.4)

We prove that for each small y_{q} > 0, these exists ϵ > 0 such that ∀z_{0} = (x_{0}, y_{0}) ∈ W_{ϵ}^{u}(q), q = (0, y_{q}), with x_{0} > 0,

v_{z_{i}}^{s} ≥ −ρ_{i} \frac{x_{i}}{y_{i}}  (10.5)

implies

v_{z_{i+1}}^{s} ≥ −ρ_{i+1} \frac{x_{i+1}}{y_{i+1}} .  (10.6)

This will prove the result of the lemma. In fact, if (10.1) is not true, then for some large n, \frac{x_{n}}{y_{n}} > 3 and v_{z_{n}}^{s} ≥ \frac{1}{3} · \frac{x_{n}}{y_{n}} > 1, contradicting to (8.4).

By Lemma 8.3, to prove that (10.5) implies (10.6), it is enough to show that

Δ_{ρ_{i}}(x_{i}, y_{i}) = (ρ_{i} − ρ_{i+1})(1 + ϕ_{i})(1 − ψ_{i})
+ ρ_{i+1}y_{i}(1 + ϕ_{i})ψ_{y}(x_{i}, y_{i}) − y_{i}(1 − ψ_{i})ϕ_{y}(x_{i}, y_{i})
− ρ_{i}ρ_{i+1}x_{i}(1 + ϕ_{i})ψ_{x}(x_{i}, y_{i}) + ρ_{i}x_{i}(1 − ψ_{i})ϕ_{x}(x_{i}, y_{i}) ≤ 0.

By (10.3) and (10.4),

ρ_{i} − ρ_{i+1} = \{ \begin{matrix}ρ̂(y_{0}) − ρ̂(y_{1}) + c_{1}, & if i = 0; \\ ρ̂(y_{i}) − ρ̂(y_{i+1}) + \frac{7}{8}c_{i+1}y_{i}ψ_{y}(0, y_{i}) − d_{i+1}, & if i > 0 & ρ_{i} > −\frac{1}{3}; \\ 0, & if i > 0 & ρ_{i} = −\frac{1}{3}, \end{matrix}

where d_{i+1} = ρ(y_{i+1}) − (\frac{a_{2}}{b_{2}} + ρ̂(y_{i+1}) − c_{i+1}) ≥ 0, and d_{i+1} = 0 except for the case ρ_{i} > −\frac{1}{3} and \frac{a_{2}}{b_{2}} − ρ̂(y_{i+1}) + c_{i+1} < \frac{1}{3}.

If ρ_{i+1} > −\frac{1}{3}, we write

ρ_{i+1}y_{i}ψ_{y}(x_{i}, y_{i})−y_{i}ϕ_{y}(x_{i}, y_{i}) = (\frac{a_{2}}{b_{2}} + ρ̂(y_{i+1}))y_{i}ψ_{y}(0, y_{i}) − y_{i}ϕ_{y}(0, y_{i})
−(c_{i+1} − d_{i+1})y_{i}ψ_{y}(0, y_{i}) + (ρ_{i+1}b_{1} − a_{1})x_{i}y_{i} + x_{i}y_{i}O(x_{i} + y_{i});

otherwise,

\begin{matrix}ρ_{i+1}y_{i}ψ_{y}(x_{i}, y_{i}) − y_{i}ϕ_{y}(x_{i}, y_{i}) \\ = & 2(ρ_{i+1}b_{2} − a_{2})y_{i}^{2} + (ρ_{i+1}b_{1} − a_{1})x_{i}y_{i} + x_{i}y_{i}O(x_{i} + y_{i}). \end{matrix}

Also,

\begin{aligned}& − ρ_{i}ρ_{i+1}x_{i}ψ_{x}(x_{i}, y_{i}) + ρ_{i}x_{i}ϕ_{x}(x_{i}, y_{i}) \\ = & − ρ_{i}(ρ_{i+1}b_{1} − a_{1})x_{i}y_{i} − 2ρ_{i}(ρ_{i+1}b_{0} − a_{0})x_{i}^{2} + x_{i}y_{i}O(x_{i} + y_{i}). \end{aligned}

Similarly as in the proof of Lemma 9.1, we use Lemma 8.4 to get Δ_{ρ_{i}}(x_{i}, y_{i}) = D_{i}(x_{i}, y_{i})+(1−ρ_{i})(ρ_{i+1}b_{1}−a_{1})x_{i}y_{i}−2ρ_{i}(ρ_{i+1}b_{0}−a_{0})x_{i}^{2}+x_{i}y_{i}O(x_{i}+y_{i}), where

D_{i}(x_{i}, y_{i}) ≤ \{ \begin{matrix}\frac{a_{2}}{b_{2}} (a_{1} − \frac{a_{2}}{b_{2}} b_{1})x_{0}y_{0} & if i = 0; \\ −\frac{1}{8}c_{i+1}y_{i}ψ_{y}(0, y_{i}) & if i > 0 & ρ_{i} > −\frac{1}{3}; \\ 2(ρ_{i+1}b_{2} − a_{2})y_{i}^{2} & if i > 0 & ρ_{i} = −\frac{1}{3}. \end{matrix}

Recall 2a_{2} < αb_{2} and b_{1} < 0 < a_{1}.

Since ρ_{0} = \frac{a_{2}}{b_{2}} + O(y_{0}) ≤ \frac{α}{2} + O(y_{0}) and ρ_{1} = ρ_{0} + O(y_{0}^{2}), we have ρ_{1}b_{1} − a_{1} = \frac{a_{2}}{b_{2}} b_{1} − a_{1} + O(y_{0}) ≤ 0 and 1 − ρ_{0} − \frac{a_{2}}{b_{2}} > 0 if y_{0} is small. So if Kx_{0} ≤ y_{0} for some large K, then Δ_{ρ_{0}}(x_{0}, y_{0}) ≤ 0.

For the case i > 0 and ρ_{i} > −\frac{1}{3}, by Lemma 10.1, we can choose ϵ small enough such that Kx_{i} ≤ y_{i} for some large K. Also, y_{i}ψ_{y}(0, y_{i}) = 2b_{2}y_{i}^{2} + O(y_{i}^{3}). So if ρ_{i+1} ≥ 0, then both y_{i}^{2} and x_{i}y_{i} terms has negative sign, and if ρ_{i+1} ≤ 0, then c_{i+1} > \frac{a_{2}}{2b_{2}} and therefore the first term can control other terms.

For the case ρ_{i} = −\frac{1}{3}, we can see that Δ_{ρ_{i}}(x_{i}, y_{i}) is equal to

\begin{aligned}& 2(−\frac{1}{3}b_{2} − a_{2})y_{i}^{2} + \frac{4}{3}(−\frac{1}{3}b_{1} − a_{1})x_{i}y_{i} + \frac{2}{3}(−\frac{1}{3}b_{0} − a_{0})x_{i}^{2} + x_{i}y_{i}O(x_{i} + y_{i}) \\ = & − \frac{2}{3}(3a_{2}y_{i}^{2} + 2a_{1}x_{i}y_{i} + a_{0}x_{i}^{2}) − \frac{2}{9}(3b_{2}y_{i}^{2} + 2b_{1}x_{i}y_{i} + b_{0}x_{i}^{2}) + x_{i}y_{i}O(x_{i} + y_{i}). \end{aligned}

By (8.2) it is less than 0. This proves the lemma.

Lemma 10.3. Let z_{0} = (x_{0}, y_{0}) with x_{0}, y_{0} > 0. If for all z = (x, y) in the stable curve that joins z̄_{0} and z_{1},

v_{z_{0}}^{s} ≤ −( \frac{a_{2}}{b_{2}} + ρ̂(y_{0}))\frac{x_{0}}{y_{0}} ,

then

x̄_{0} ≤ x_{0} + Cx_{0}^{2}

for some constant C = C(y_{q}) > 0, where x̄ satisfies z̄ = (x̄, ȳ) = [z, fz].

Proof. Using a similar way as in the proof of Lemma 9.3, we can get

\frac{x̄_{0}}{x_{0}} ≤ exp \frac{1}{b_{2}} [b_{2} log(1 + ϕ_{0}) + a_{2} log(1 − ψ_{0}) − b_{2} ∫_{y_{1}}^{y_{0}} \frac{ρ̂(y)}{y}dy](\frac{y_{0}}{ȳ_{0}} )^{\frac{a_{2}}{b_{2}}}
· exp ∫ y_{0} ; ȳ_{0} \frac{ρ̂(y)}{y}dy ≤ 1 + O(x_{0}).

Therefore the result follows.

Proof of Theorem B (II).

First we assume that b_{1} < 0 < a_{1}. By Lemma 10.3 and 8.1.ii), Δ^{++} = ∞, where Δ^{++} is defined in (7.6). Therefore by Proposition 7.8 f has an infinite SBR measure.

By (1.3) and (1.4), the signs of a_{1} and b_{1} change under the transformation x → −x and y → −y respectively. So under the assumption of Theorem B (II), we can always find a suitable coordinate system such that b_{1} < 0 < a_{1}.

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Department of Mathematics, University of Maryland, College Park, MD 20742

   E-mail address: huyi@math.umd.edu